{"id":2488,"date":"2022-12-26T23:12:04","date_gmt":"2022-12-26T22:12:04","guid":{"rendered":"https:\/\/ff.mhrooz.xyz\/?p=2488"},"modified":"2023-01-10T22:19:29","modified_gmt":"2023-01-10T21:19:29","slug":"recap_linear_algebra","status":"publish","type":"post","link":"https:\/\/blog.mhrooz.xyz\/index.php\/2022\/12\/26\/recap_linear_algebra\/","title":{"rendered":"Recap: Linear Algebra"},"content":{"rendered":"\n<!doctype html>\n<html>\n<head>\n<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>\n\n<style type='text\/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: \"Lucida Console\",Consolas,\"Courier\",monospace; --title-bar-height: 20px; }\n.mac-os-11 { --title-bar-height: 28px; }\nhtml { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: \"Helvetica Neue\", Helvetica, Arial, sans-serif; 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padding-right: 0;} #write {padding:0;}}\n<\/style><title>Chapter 4 Determinants<\/title>\n<\/head>\n<body class='typora-export'><div class='typora-export-content'>\n<div id='write'  class=''><p>&nbsp;<\/p><h1 id='chapter-4-determinants'><span>Chapter 4 Determinants<\/span><\/h1><h2 id='1-uses-of-determinants'><span>1. Uses of Determinants<\/span><\/h2><ul><li><span>Invertibility<\/span><\/li><\/ul><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.103ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3139.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-24-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-24-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-24-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-24-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-24-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-24-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1028,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-24-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1583.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-24-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2639.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-24-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\begin{vmatrix}\nA\n\\end{vmatrix}  = 0<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is singular. If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.103ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3139.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-26-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-26-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-26-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z\"><\/path><path id=\"MJX-26-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-26-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-26-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1028,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-26-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1583.8,0)\"><use data-c=\"2260\" xlink:href=\"#MJX-26-TEX-N-2260\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2639.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-26-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>\u2260<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\begin{vmatrix}\nA\n\\end{vmatrix} \u2260 0<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is invertible.<\/span><\/p><ul><li><span>Volume<\/span><\/li><\/ul><p><span>The determinant of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> equals the volume of a box in n-dimensional space. The edges of the box come from the rows of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span>. The columns of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> would give an entirely different box with the same volume.<\/span><\/p><ul><li><span>Pivots<\/span><\/li><\/ul><p><span>The determinant gives a formula for each pivot. From the formula <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"39.881ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 17627.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-31-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 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stretchy=\"false\">(<\/mo><mi>p<\/mi><mi>r<\/mi><mi>o<\/mi><mi>d<\/mi><mi>u<\/mi><mi>c<\/mi><mi>t<\/mi><mtext>&nbsp;<\/mtext><mi>o<\/mi><mi>f<\/mi><mtext>&nbsp;<\/mtext><mi>t<\/mi><mi>h<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>p<\/mi><mi>i<\/mi><mi>v<\/mi><mi>o<\/mi><mi>t<\/mi><mi>s<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">determinant = \u00b1 (product\\ of\\ the\\ pivots)<\/script><span>, it follows that regardless of the order of elimination, the product of the pivots remains the same apart from the sign.<\/span><\/p><ul><li><span>Dependence<\/span><\/li><\/ul><p><span>The determinant measures the dependence of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.9ex\" height=\"1.912ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 2165.7 844.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" 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data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-32-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1736.7,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-32-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>A<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A^{-1}b<\/script><span> on each element of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.971ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 429 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-33-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-33-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">b<\/script><span>.<\/span><\/p><h2 id='2-properties-of-determinants'><span>2. Properties of Determinants<\/span><\/h2><ul><li><span>The determinant of the identity matrix is 1.<\/span><\/li><li><span>The determinant changes sign when two rows are exchanged.<\/span><\/li><\/ul><p><em><span>The determinant of every permutation matrix is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.866ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3918.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-34-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-34-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-34-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-34-TEX-N-B1\" d=\"M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z\"><\/path><path id=\"MJX-34-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-34-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-34-TEX-I-1D443\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1029,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-34-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1584.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-34-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2640.6,0)\"><use data-c=\"B1\" xlink:href=\"#MJX-34-TEX-N-B1\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3418.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-34-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>P<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mo>\u00b1<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\begin{vmatrix}\nP\n\\end{vmatrix} = \u00b11<\/script><span>.<\/span><\/em><\/p><ul><li><span>The determinant depends linearly on the first row.<\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n376\" cid=\"n376\" mdtype=\"math_block\" data-math-tag-before=\"0\" 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292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-2-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-2-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path id=\"MJX-2-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-2-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-2-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-2-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018137,-0.018137) translate(0, -1454.5)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1454.5) matrix(1 0 0 -1 0 0) scale(55.14)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"7515.9 -1454.5 1 2409\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mrow\"><g 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data-c=\"2223\" xlink:href=\"#MJX-2-TEX-S4-2223\" transform=\"scale(1,3.617)\"><\/use><\/svg><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6859.6,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-2-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(7915.4,0)\"><g data-mml-node=\"mo\"><svg width=\"278\" height=\"2400\" y=\"-950\" x=\"27.5\" viewBox=\"0 -297.3 278 2400\"><use data-c=\"2223\" xlink:href=\"#MJX-2-TEX-S4-2223\" transform=\"scale(1,3.604)\"><\/use><\/svg><\/g><g data-mml-node=\"mtable\" transform=\"translate(333,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,700)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-2-TEX-I-1D44E\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1574.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-2-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-700)\"><g data-mml-node=\"mtd\" transform=\"translate(48,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-2-TEX-I-1D450\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1529,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-2-TEX-I-1D451\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2382,0)\"><svg width=\"278\" height=\"2400\" y=\"-950\" x=\"27.5\" viewBox=\"0 -297.3 278 2400\"><use data-c=\"2223\" xlink:href=\"#MJX-2-TEX-S4-2223\" transform=\"scale(1,3.604)\"><\/use><\/svg><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10852.6,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-2-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(11852.8,0)\"><g data-mml-node=\"mo\"><svg width=\"278\" height=\"2409\" y=\"-954.5\" x=\"27.5\" viewBox=\"0 -298.4 278 2409\"><use data-c=\"2223\" xlink:href=\"#MJX-2-TEX-S4-2223\" transform=\"scale(1,3.617)\"><\/use><\/svg><\/g><g data-mml-node=\"mtable\" transform=\"translate(333,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,695.5)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-2-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(562,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-2-TEX-V-2032\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1806.5,0)\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-2-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(462,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-2-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-704.5)\"><g data-mml-node=\"mtd\" transform=\"translate(186.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-2-TEX-I-1D450\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1899.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-2-TEX-I-1D451\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2845.9,0)\"><svg width=\"278\" height=\"2409\" y=\"-954.5\" x=\"27.5\" viewBox=\"0 -298.4 278 2409\"><use data-c=\"2223\" xlink:href=\"#MJX-2-TEX-S4-2223\" transform=\"scale(1,3.617)\"><\/use><\/svg><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1454.5 1 2409\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>a<\/mi><mo>+<\/mo><msup><mi>a<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mtd><mtd><mi>b<\/mi><mo>+<\/mo><msup><mi>b<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>a<\/mi><\/mtd><mtd><mi>b<\/mi><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msup><mi>a<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mtd><mtd><msup><mi>b<\/mi><mo 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transform=\"scale(1,3.617)\"><\/use><\/svg><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1454.5 1 2409\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>t<\/mi><mi>a<\/mi><\/mtd><mtd><mi>t<\/mi><mi>b<\/mi><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mi>t<\/mi><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>a<\/mi><\/mtd><mtd><msup><mi>b<\/mi><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><ul><li><span>If two rows of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> are equal, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.103ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3139.6 1000\" 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140 56 153Z\"><\/path><path id=\"MJX-36-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-36-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-36-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1028,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-36-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1583.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-36-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2639.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-36-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\begin{vmatrix}\nA\n\\end{vmatrix} = 0<\/script><span>.<\/span><\/li><li><span>Subtracting a multiple of one row from another row leaves the same determinant.<\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block 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transform=\"translate(1574.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-4-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-700)\"><g data-mml-node=\"mtd\" transform=\"translate(48,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-4-TEX-I-1D450\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(1529,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-4-TEX-I-1D451\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2382,0)\"><svg width=\"278\" height=\"2400\" y=\"-950\" x=\"27.5\" viewBox=\"0 -297.3 278 2400\"><use data-c=\"2223\" xlink:href=\"#MJX-4-TEX-S4-2223\" transform=\"scale(1,3.604)\"><\/use><\/svg><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1450 1 2400\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>a<\/mi><mo>\u2212<\/mo><mi>l<\/mi><mi>c<\/mi><\/mtd><mtd><mi>b<\/mi><mo>\u2212<\/mo><mi>l<\/mi><mi>d<\/mi><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>a<\/mi><\/mtd><mtd><mi>b<\/mi><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><p><em><span>The usual elimination steps do not affect the determinant.<\/span><\/em><\/p><ul><li><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> has a row of zeros, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.103ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3139.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-38-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-38-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-38-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-38-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-38-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-38-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1028,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-38-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1583.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-38-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2639.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-38-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\begin{vmatrix}\nA\n\\end{vmatrix}=0<\/script><span> .<\/span><\/li><li><span>*If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is triangular, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.955ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1306 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-40-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-40-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-40-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-40-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1028,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-40-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\begin{vmatrix}\nA\n\\end{vmatrix}<\/script><span> is the product of the diagonal entries. <\/span><\/li><li><span>The determinant of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.414ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 1509 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-41-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-41-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-41-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-41-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>B<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AB<\/script><span> is the product of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.744ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 2097 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-42-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-42-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 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0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-42-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(520,0)\"><use data-c=\"1D452\" xlink:href=\"#MJX-42-TEX-I-1D452\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(986,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-42-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1347,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-42-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math 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616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-43-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><path id=\"MJX-43-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-43-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g 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0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-5-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-5-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 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columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>B<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><mi>B<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n395\" cid=\"n395\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" 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transform=\"translate(2014.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-6-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2570.5,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-6-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(3626.2,0)\"><g data-mml-node=\"mn\" transform=\"translate(623,676)\"><use data-c=\"31\" xlink:href=\"#MJX-6-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-6-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-6-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1028,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-6-TEX-N-7C\"><\/use><\/g><\/g><rect width=\"1506\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1401 1 2302\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,59)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msup><mi>A<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><span>The transpose of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> has the same determinant as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> itself.<\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n399\" cid=\"n399\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 13.861ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.47ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.669ex; min-width: 13.861ex;\"><defs><path id=\"MJX-7-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-7-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-7-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-7-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018137,-0.018137) translate(0, -795.9)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 795.9) matrix(1 0 0 -1 0 0) scale(55.14)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"2263.2 -795.9 1 1091.7\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-7-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-7-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1028,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-7-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1583.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-7-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(2639.6,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-7-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,-45.9)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-7-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(783,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-7-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1608.8,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-7-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -795.9 1 1091.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msup><mi>A<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>T<\/mi><\/mrow><\/msup><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><em><span>LDU<\/span><\/em><span> factorization.<\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n403\" cid=\"n403\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 15.182ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 15.182ex;\"><defs><path id=\"MJX-8-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-8-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-8-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-8-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-8-TEX-I-1D437\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path id=\"MJX-8-TEX-I-1D448\" d=\"M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 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transform=\"translate(751,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-8-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1778.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-8-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2834.6,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-8-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3515.6,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-8-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4343.6,0)\"><use data-c=\"1D448\" xlink:href=\"#MJX-8-TEX-I-1D448\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>P<\/mi><mi>A<\/mi><mo>=<\/mo><mi>L<\/mi><mi>D<\/mi><mi>U<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n404\" cid=\"n404\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 41.135ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 41.135ex;\"><defs><path id=\"MJX-9-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-9-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-9-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-9-TEX-N-B1\" d=\"M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 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display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mo>\u00b1<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>P<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mo>\u00b1<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>L<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>D<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>U<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mo>\u00b1<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>D<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h2 id='3-applications-of-determinants'><span>3. Applications of Determinants<\/span><\/h2><ul><li><span>Computation of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.929ex\" height=\"1.887ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 1736.7 833.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-103-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path 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xlink:href=\"#MJX-103-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>A<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A^{-1}<\/script><span>.<\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n409\" cid=\"n409\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 52.53ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.9ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msup><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>a<\/mi><\/mtd><mtd><mi>b<\/mi><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>=<\/mo><mfrac><mn>1<\/mn><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><\/mfrac><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mn>11<\/mn><\/mrow><\/msub><\/mtd><mtd><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mn>12<\/mn><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mn>21<\/mn><\/mrow><\/msub><\/mtd><mtd><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mn>22<\/mn><\/mrow><\/msub><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>a<\/mi><mi>d<\/mi><mo>\u2212<\/mo><mi>b<\/mi><mi>c<\/mi><\/mrow><\/mfrac><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>d<\/mi><\/mtd><mtd><mo>\u2212<\/mo><mi>b<\/mi><\/mtd><\/mtr><mtr><mtd><mo>\u2212<\/mo><mi>c<\/mi><\/mtd><mtd><mi>a<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><span>Cramer&#39;s Rule.<\/span><\/li><\/ul><h1 id='chapter-5-eigenvalues-and-eigenvectors'><span>Chapter 5 Eigenvalues and Eigenvectors<\/span><\/h1><h2 id='1-introduction'><span>1. Introduction<\/span><\/h2><p><span>The eigenvalues are the most important feature of practically any dynamic system. Until now we have focused on the problem <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.979ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3084.6 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-47-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-47-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-47-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-47-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-47-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-47-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1599.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-47-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2655.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-47-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>x<\/mi><mo>=<\/mo><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Ax=b<\/script><span>, now we consider the new problem <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.621ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3810.6 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-59-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-59-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-59-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-59-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-59-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-59-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1599.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-59-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2655.6,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-59-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3238.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-59-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>x<\/mi><mo>=<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Ax=\\lambda x<\/script><span>. This will still be solved by simplifying a matrix, but the basic step is no longer to subtract a multiple of one row from another: Elimination changes the eigenvalues. <\/span><\/p><h3 id='1-solution-of-\\na\\nx\\n\\n\u03bb\\nx'><span>(1) Solution of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.621ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3810.6 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-59-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-59-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-59-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-59-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-59-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-59-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1599.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-59-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2655.6,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-59-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3238.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-59-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>x<\/mi><mo>=<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Ax=\\lambda x<\/script><\/h3><p><span>This is a nonlinear equation since both <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.319ex\" height=\"1.597ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 583 706\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.027ex;\"><defs><path id=\"MJX-64-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-64-TEX-I-1D706\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda<\/script><span> are unknown. We first discover <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.319ex\" height=\"1.597ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 583 706\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.027ex;\"><defs><path id=\"MJX-64-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-64-TEX-I-1D706\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda<\/script><span>:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n418\" cid=\"n418\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 18.373ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 18.373ex;\"><defs><path id=\"MJX-11-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-11-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-11-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-11-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-11-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-11-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-11-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-11-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-11-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-11-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018137,-0.018137) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(55.14)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3260.5 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-11-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-11-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1361.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-11-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2361.4,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-11-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2944.4,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-11-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3448.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-11-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3837.4,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-11-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4687.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-11-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5743,0)\"><use data-c=\"30\" xlink:href=\"#MJX-11-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-11-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><blockquote><p><span>The vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-65-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-65-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span> is in the nullspace of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.922ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3059.4 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-63-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-63-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-63-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-63-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-63-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(972.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-63-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.4,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-63-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2555.4,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-63-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A-\\lambda I<\/script><span>.<\/span><\/p><p><span>The number <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.319ex\" height=\"1.597ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 583 706\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.027ex;\"><defs><path id=\"MJX-64-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-64-TEX-I-1D706\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda<\/script><span> is chosen so that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.922ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3059.4 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-63-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-63-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-63-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-63-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-63-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(972.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-63-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.4,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-63-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2555.4,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-63-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A-\\lambda I<\/script><span> has a nullspace.<\/span><\/p><\/blockquote><p><span>Of course, every matrix has a nullspace, but we want a nonzero eigenvector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-65-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-65-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span>. The vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.442ex\" height=\"1.692ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 2405.6 748\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-58-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-58-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-58-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-58-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(849.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-58-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1905.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-58-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x=0<\/script><span> satisfies  <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.621ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3810.6 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-59-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-59-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-59-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-59-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-59-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-59-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1599.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-59-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2655.6,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-59-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3238.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-59-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>x<\/mi><mo>=<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Ax=\\lambda x<\/script><span>, but it is useless in solving differential equations. We are interested only in those particular values <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.319ex\" height=\"1.597ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 583 706\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.027ex;\"><defs><path id=\"MJX-64-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-64-TEX-I-1D706\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda<\/script><span> for which there is a nonzero eigenvector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-65-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-65-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span>. That is, the nullspace of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.922ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3059.4 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-63-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-63-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-63-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-63-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-63-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(972.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-63-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.4,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-63-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2555.4,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-63-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A-\\lambda I<\/script><span> must contain vectors other than 0. In short, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.922ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3059.4 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-63-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-63-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-63-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-63-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-63-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(972.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-63-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.4,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-63-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2555.4,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-63-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A-\\lambda I<\/script><span> must be singular.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n423\" cid=\"n423\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 16.102ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 16.102ex;\"><defs><path id=\"MJX-12-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-12-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-12-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-12-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-12-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-12-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-12-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018137,-0.018137) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(55.14)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"2758.5 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-12-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtable\" transform=\"translate(278,0)\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-12-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(972.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-12-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.4,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-12-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2555.4,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-12-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3405.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-12-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3961.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-12-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5017,0)\"><use data-c=\"30\" xlink:href=\"#MJX-12-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Each <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.319ex\" height=\"1.597ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 583 706\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.027ex;\"><defs><path id=\"MJX-64-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-64-TEX-I-1D706\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda<\/script><span> is associated with eigenvectors <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-65-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use 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29.614ex;\"><defs><path id=\"MJX-13-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-13-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-13-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path 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transform=\"translate(10917.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-13-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>I<\/mi><mo stretchy=\"false\">)<\/mo><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><mtext>&nbsp;<\/mtext><mi>o<\/mi><mi>r<\/mi><mtext>&nbsp;<\/mtext><mi>A<\/mi><mi>x<\/mi><mo>=<\/mo><mi>\u03bb<\/mi><mi>x<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='2-checks-on-eigenvalues'><span>(2) Checks on Eigenvalues<\/span><\/h3><p><span>The <\/span><em><span>sum<\/span><\/em><span> of eigenvalues equals the <\/span><em><span>sum<\/span><\/em><span> of the diagonal entries:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n428\" cid=\"n428\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 42.318ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 42.318ex;\"><defs><path id=\"MJX-14-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 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63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-14-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-14-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-14-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-14-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><path id=\"MJX-14-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-14-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 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transform=\"translate(13697.6,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-14-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14148.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-14-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14677.6,0)\"><use data-c=\"1D450\" xlink:href=\"#MJX-14-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15110.6,0)\"><use data-c=\"1D452\" xlink:href=\"#MJX-14-TEX-I-1D452\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15576.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-14-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15965.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-14-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16715.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-14-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>\u03bb<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>+<\/mo><msub><mi>\u03bb<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><msub><mi>a<\/mi><mrow data-mjx-texclass=\"ORD\"><mn>11<\/mn><\/mrow><\/msub><mo>+<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>+<\/mo><msub><mi>a<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>n<\/mi><mi>n<\/mi><\/mrow><\/msub><mo>=<\/mo><mi>t<\/mi><mi>r<\/mi><mi>a<\/mi><mi>c<\/mi><mi>e<\/mi><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Furthermore, the product of eigenvalues equals the determinant of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span>.<\/span><\/p><p><span>From example** of eigenvalues, the diagonal entries and the eigenvalues are the <\/span><em><span>same<\/span><\/em><span> only in triangular matrices. Normally, they are completely different.<\/span><\/p><h3 id='3-examples'><span>(3) Examples<\/span><\/h3><ul><li><span>Everything is clear when <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is a diagonal matrix:<\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n435\" cid=\"n435\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 41.495ex; position: relative;\"><svg 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rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>2<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> acts like a multiple of the identity on each eigenvector:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n437\" cid=\"n437\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" 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394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-16-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 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3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-16-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018137,-0.018137) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(55.14)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5814.2 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-16-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(750,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-16-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-16-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2036.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-16-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3092.1,0)\"><use data-c=\"33\" xlink:href=\"#MJX-16-TEX-N-33\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3592.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-16-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-16-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(4600.7,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-16-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4850.7,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-16-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5379.7,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-16-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5979.7,0)\"><use data-c=\"1D451\" xlink:href=\"#MJX-16-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(6499.7,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-16-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6749.7,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-16-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7499.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-16-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-16-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8786,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-16-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9841.8,0)\"><use data-c=\"32\" xlink:href=\"#MJX-16-TEX-N-32\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(10341.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-16-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-16-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11350.3,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-16-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>A<\/mi><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>3<\/mn><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mtext>&nbsp;<\/mtext><mi>a<\/mi><mi>n<\/mi><mi>d<\/mi><mtext>&nbsp;<\/mtext><mi>A<\/mi><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>2<\/mn><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo>.<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The action of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is determined by its eigenvectors and eigenvalues.<\/span><\/p><ul><li><span>The eigenvalues of a projection matrix are 1 or 0.<\/span><\/li><\/ul><div 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displaystyle=\"true\"><mtr><mtd><mi>P<\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mtd><mtd><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mtd><\/mtr><mtr><mtd><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mtd><mtd><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mtext>&nbsp;<\/mtext><mi>h<\/mi><mi>a<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><msub><mi>\u03bb<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mn>1<\/mn><mtext>&nbsp;<\/mtext><mi>w<\/mi><mi>i<\/mi><mi>t<\/mi><mi>h<\/mi><mtext>&nbsp;<\/mtext><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>,<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>\u03bb<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>0<\/mn><mtext>&nbsp;<\/mtext><mi>w<\/mi><mi>i<\/mi><mi>t<\/mi><mi>h<\/mi><mtext>&nbsp;<\/mtext><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mo>\u2212<\/mo><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>.<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>A zero eigenvalue signals that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is singular (not invertible); its determinant is zero. Invertible matrices have all <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.467ex\" height=\"2.106ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 2416.6 931\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.486ex;\"><defs><path id=\"MJX-71-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-71-TEX-N-2260\" d=\"M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 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xlink:href=\"#MJX-71-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><mo>\u2260<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda \u22600<\/script><span>.<\/span><\/p><ul><li><span>**The eigenvalues are on the main diagonal <\/span><em><span>when <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is triangular<\/span><\/em><span>:   <\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n447\" cid=\"n447\" mdtype=\"math_block\" 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xlink:href=\"#MJX-19-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(25610.3,0)\"><use data-c=\"1D706\" xlink:href=\"#MJX-19-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(26193.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-19-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2359.6 1 4219.2\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>A<\/mi><mo>\u2212<\/mo><mi>\u03bb<\/mi><mi>I<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">|<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><\/mtd><mtd><mn>4<\/mn><\/mtd><mtd><mn>5<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mo>\u2212<\/mo><mi>\u03bb<\/mi><\/mtd><mtd><mn>6<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>\u2212<\/mo><mi>\u03bb<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">|<\/mo><\/mrow><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>\u2212<\/mo><mi>\u03bb<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>This follows from property* of determinants. <\/span><\/p><p><span>The eigenvalues are <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.467ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2416.6 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-73-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-73-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-73-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-73-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(860.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-73-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1916.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-73-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda=1<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.131ex\" height=\"2.736ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.2 2710.1 1209.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.781ex;\"><defs><path id=\"MJX-74-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-74-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-74-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-74-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-74-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(860.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-74-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(1916.6,0)\"><g data-mml-node=\"mn\" transform=\"translate(220,394) scale(0.707)\"><use data-c=\"33\" xlink:href=\"#MJX-74-TEX-N-33\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(220,-345) scale(0.707)\"><use data-c=\"34\" xlink:href=\"#MJX-74-TEX-N-34\"><\/use><\/g><rect width=\"553.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>4<\/mn><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda=\\frac{3}{4} <\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.131ex\" height=\"2.737ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -864.9 2710.1 1209.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.781ex;\"><defs><path id=\"MJX-75-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-75-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-75-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-75-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-75-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(860.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-75-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(1916.6,0)\"><g data-mml-node=\"mn\" transform=\"translate(220,394) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-75-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(220,-345) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-75-TEX-N-32\"><\/use><\/g><rect width=\"553.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda=\\frac{1}{2}<\/script><span>, which are the diagonal entries of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span>.<\/span><\/p><h2 id='2-diagonalization-of-a-matrix'><span>2. Diagonalization of a Matrix<\/span><\/h2><h3 id='1-eigenvectors-diagonalize-a-matrix'><span>(1) Eigenvectors <\/span><em><span>Diagonalize<\/span><\/em><span> a Matrix<\/span><\/h3><blockquote><p><span>Suppose the <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.481ex\" height=\"1.136ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -491 2422.4 502\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-77-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-77-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-77-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(822.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-77-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1822.4,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-77-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n\\times n<\/script><span>  matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> has <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-79-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-79-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span> linearly independent eigenvectors. If these vectors are the columns of a matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-93-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-93-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span>, then <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.964ex\" height=\"1.937ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 3078.3 855.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-81-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-81-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-81-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-81-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-81-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(729.6,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-81-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-81-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(1683.3,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-81-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2433.3,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-81-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>S<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>A<\/mi><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S^{-1}AS<\/script><span> is a diagonal matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.57ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 694 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-94-TEX-N-39B\" d=\"M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"39B\" xlink:href=\"#MJX-94-TEX-N-39B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u039b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Lambda<\/script><span>. The eigenvalues of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> are on the diagonal of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.57ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 694 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-94-TEX-N-39B\" d=\"M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"39B\" xlink:href=\"#MJX-94-TEX-N-39B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u039b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Lambda<\/script><span>:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n455\" cid=\"n455\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 36.709ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"11.765ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -5.317ex; min-width: 36.709ex;\"><defs><path id=\"MJX-20-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 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transform=\"translate(0,-700)\"><g data-mml-node=\"mtd\" transform=\"translate(509.8,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(2529.3,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4039.1,0)\"><g data-mml-node=\"mo\"><use data-c=\"2E\" xlink:href=\"#MJX-20-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(444.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-20-TEX-N-2E\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6306.9,0)\"><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-2100)\"><g data-mml-node=\"mtd\" transform=\"translate(509.8,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(2529.3,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4400.4,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(5761.8,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-20-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-20-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7519,0)\"><use data-c=\"23A4\" xlink:href=\"#MJX-20-TEX-S4-23A4\" transform=\"translate(0,1696)\"><\/use><use data-c=\"23A6\" xlink:href=\"#MJX-20-TEX-S4-23A6\" transform=\"translate(0,-1706)\"><\/use><svg width=\"667\" height=\"1802\" y=\"-651\" x=\"0\" viewBox=\"0 450.5 667 1802\"><use data-c=\"23A5\" xlink:href=\"#MJX-20-TEX-S4-23A5\" transform=\"scale(1,4.49)\"><\/use><\/svg><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2850 1 5200\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><msup><mi>S<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>A<\/mi><mi>S<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u039b<\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mi>\u03bb<\/mi><mn>1<\/mn><\/msub><\/mtd><mtd><\/mtd><mtd><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msub><mi>\u03bb<\/mi><mn>2<\/mn><\/msub><\/mtd><mtd><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><\/mtd><mtd><mo>.<\/mo><mo>.<\/mo><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><\/mtd><mtd><\/mtd><mtd><msub><mi>\u03bb<\/mi><mi>n<\/mi><\/msub><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We call <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-93-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-93-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span> the &quot;eigenvector matrix&quot; and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.57ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 694 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-94-TEX-N-39B\" d=\"M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"39B\" xlink:href=\"#MJX-94-TEX-N-39B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u039b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Lambda<\/script><span> the &quot;eigenvalue matrix&quot;.<\/span><\/p><\/blockquote><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.787ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 3000 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-87-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-87-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-87-TEX-I-1D45C\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><path id=\"MJX-87-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 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transform=\"translate(751,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-87-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1202,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-87-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1687,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-87-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2172,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-87-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2722,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-87-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>P<\/mi><mi>r<\/mi><mi>o<\/mi><mi>o<\/mi><mi>f<\/mi><mo>.<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Proof.<\/script><span> <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block 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stretchy=\"false\">|<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>\u03bb<\/mi><mn>1<\/mn><\/msub><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mtd><mtd><msub><mi>\u03bb<\/mi><mn>2<\/mn><\/msub><msub><mi>x<\/mi><mn>2<\/mn><\/msub><\/mtd><mtd><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><\/mtd><mtd><msub><mi>\u03bb<\/mi><mi>n<\/mi><\/msub><msub><mi>x<\/mi><mi>n<\/mi><\/msub><\/mtd><\/mtr><mtr><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We split the matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.156ex\" height=\"1.67ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 1395 738\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-88-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-88-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-88-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-88-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AS<\/script><span> into product <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.029ex\" height=\"1.67ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 1339 738\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-89-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-89-TEX-N-39B\" d=\"M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-89-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(645,0)\"><use data-c=\"39B\" xlink:href=\"#MJX-89-TEX-N-39B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><mi mathvariant=\"normal\">\u039b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S\\Lambda<\/script><span>:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n460\" cid=\"n460\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 71.518ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"11.765ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -5.317ex; min-width: 71.518ex;\"><defs><path id=\"MJX-22-TEX-S4-23A1\" d=\"M319 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52T207 57T213 61V578Z\"><\/path><path id=\"MJX-22-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-22-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-22-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-22-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-22-TEX-S4-23A4\" d=\"M0 1070V1154H347V-645H263V1070H0Z\"><\/path><path id=\"MJX-22-TEX-S4-23A6\" d=\"M263 -560V1155H347V-644H0V-560H263Z\"><\/path><path id=\"MJX-22-TEX-S4-23A5\" d=\"M263 0V602H347V0H263Z\"><\/path><path id=\"MJX-22-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.018137,-0.018137) translate(0, -2850)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 2850) matrix(1 0 0 -1 0 0) scale(55.14)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"15005.5 -2850 1 5200\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mrow\"><g data-mml-node=\"mo\"><use data-c=\"23A1\" xlink:href=\"#MJX-22-TEX-S4-23A1\" transform=\"translate(0,996)\"><\/use><use data-c=\"23A3\" xlink:href=\"#MJX-22-TEX-S4-23A3\" transform=\"translate(0,-1006)\"><\/use><svg width=\"667\" height=\"402\" y=\"49\" x=\"0\" viewBox=\"0 100.5 667 402\"><use data-c=\"23A2\" xlink:href=\"#MJX-22-TEX-S4-23A2\" transform=\"scale(1,1.002)\"><\/use><\/svg><\/g><g data-mml-node=\"mtable\" transform=\"translate(667,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,1400)\"><g data-mml-node=\"mtd\" transform=\"translate(875.1,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3903.2,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6639.9,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(9169.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" 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data-mml-node=\"msub\" transform=\"translate(1019.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-22-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-22-TEX-N-32\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6056.2,0)\"><g data-mml-node=\"mo\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(444.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(889.3,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(8223.5,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-22-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-22-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(1090.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-22-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-22-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1400)\"><g data-mml-node=\"mtd\" transform=\"translate(875.1,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3903.2,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6639.9,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(9169.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11060.1,0)\"><use data-c=\"23A4\" xlink:href=\"#MJX-22-TEX-S4-23A4\" transform=\"translate(0,996)\"><\/use><use data-c=\"23A6\" xlink:href=\"#MJX-22-TEX-S4-23A6\" transform=\"translate(0,-1006)\"><\/use><svg width=\"667\" height=\"402\" y=\"49\" x=\"0\" viewBox=\"0 100.5 667 402\"><use data-c=\"23A5\" xlink:href=\"#MJX-22-TEX-S4-23A5\" transform=\"scale(1,1.002)\"><\/use><\/svg><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12004.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-22-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(13060.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"23A1\" xlink:href=\"#MJX-22-TEX-S4-23A1\" transform=\"translate(0,996)\"><\/use><use data-c=\"23A3\" xlink:href=\"#MJX-22-TEX-S4-23A3\" transform=\"translate(0,-1006)\"><\/use><svg width=\"667\" height=\"402\" y=\"49\" x=\"0\" viewBox=\"0 100.5 667 402\"><use data-c=\"23A2\" xlink:href=\"#MJX-22-TEX-S4-23A2\" transform=\"scale(1,1.002)\"><\/use><\/svg><\/g><g data-mml-node=\"mtable\" transform=\"translate(667,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,1400)\"><g data-mml-node=\"mtd\" transform=\"translate(365.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(2373.8,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4600.8,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(6585.1,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-22-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-22-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(2008.6,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-22-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-22-TEX-N-32\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4017.1,0)\"><g data-mml-node=\"mo\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(444.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(889.3,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6184.4,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-22-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-22-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1400)\"><g data-mml-node=\"mtd\" transform=\"translate(365.3,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(2373.8,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4600.8,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(6585.1,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-22-TEX-N-7C\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7930.7,0)\"><use data-c=\"23A4\" xlink:href=\"#MJX-22-TEX-S4-23A4\" transform=\"translate(0,996)\"><\/use><use data-c=\"23A6\" xlink:href=\"#MJX-22-TEX-S4-23A6\" transform=\"translate(0,-1006)\"><\/use><svg width=\"667\" height=\"402\" y=\"49\" x=\"0\" viewBox=\"0 100.5 667 402\"><use data-c=\"23A5\" xlink:href=\"#MJX-22-TEX-S4-23A5\" transform=\"scale(1,1.002)\"><\/use><\/svg><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(21825,0)\"><g data-mml-node=\"mo\"><use data-c=\"23A1\" xlink:href=\"#MJX-22-TEX-S4-23A1\" transform=\"translate(0,1696)\"><\/use><use data-c=\"23A3\" xlink:href=\"#MJX-22-TEX-S4-23A3\" transform=\"translate(0,-1706)\"><\/use><svg width=\"667\" height=\"1802\" y=\"-651\" x=\"0\" viewBox=\"0 450.5 667 1802\"><use data-c=\"23A2\" xlink:href=\"#MJX-22-TEX-S4-23A2\" transform=\"scale(1,4.49)\"><\/use><\/svg><\/g><g data-mml-node=\"mtable\" transform=\"translate(667,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,2100)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-22-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-22-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(2529.3,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4400.4,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(6306.9,0)\"><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,700)\"><g data-mml-node=\"mtd\" transform=\"translate(509.8,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(2019.6,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-22-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-22-TEX-N-32\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(4400.4,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(6306.9,0)\"><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-700)\"><g data-mml-node=\"mtd\" transform=\"translate(509.8,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(2529.3,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4039.1,0)\"><g data-mml-node=\"mo\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(444.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-22-TEX-N-2E\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(6306.9,0)\"><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-2100)\"><g data-mml-node=\"mtd\" transform=\"translate(509.8,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(2529.3,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(4400.4,0)\"><\/g><g data-mml-node=\"mtd\" transform=\"translate(5761.8,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-22-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-22-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7519,0)\"><use data-c=\"23A4\" xlink:href=\"#MJX-22-TEX-S4-23A4\" transform=\"translate(0,1696)\"><\/use><use data-c=\"23A6\" xlink:href=\"#MJX-22-TEX-S4-23A6\" transform=\"translate(0,-1706)\"><\/use><svg width=\"667\" height=\"1802\" y=\"-651\" x=\"0\" viewBox=\"0 450.5 667 1802\"><use data-c=\"23A5\" xlink:href=\"#MJX-22-TEX-S4-23A5\" transform=\"scale(1,4.49)\"><\/use><\/svg><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2850 1 5200\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>\u03bb<\/mi><mn>1<\/mn><\/msub><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mtd><mtd><msub><mi>\u03bb<\/mi><mn>2<\/mn><\/msub><msub><mi>x<\/mi><mn>2<\/mn><\/msub><\/mtd><mtd><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><\/mtd><mtd><msub><mi>\u03bb<\/mi><mi>n<\/mi><\/msub><msub><mi>x<\/mi><mi>n<\/mi><\/msub><\/mtd><\/mtr><mtr><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mtd><mtd><msub><mi>x<\/mi><mn>2<\/mn><\/msub><\/mtd><mtd><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><\/mtd><mtd><msub><mi>x<\/mi><mi>n<\/mi><\/msub><\/mtd><\/mtr><mtr><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><mtd><\/mtd><mtd><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mi>\u03bb<\/mi><mn>1<\/mn><\/msub><\/mtd><mtd><\/mtd><mtd><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msub><mi>\u03bb<\/mi><mn>2<\/mn><\/msub><\/mtd><mtd><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><\/mtd><mtd><mo>.<\/mo><mo>.<\/mo><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><\/mtd><mtd><\/mtd><mtd><msub><mi>\u03bb<\/mi><mi>n<\/mi><\/msub><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Therefore, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.203ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 4067.6 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-90-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-90-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-90-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-90-TEX-N-39B\" d=\"M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-90-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-90-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1672.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-90-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2728.6,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-90-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3373.6,0)\"><use data-c=\"39B\" xlink:href=\"#MJX-90-TEX-N-39B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>S<\/mi><mo>=<\/mo><mi>S<\/mi><mi mathvariant=\"normal\">\u039b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AS=S\\Lambda<\/script><span>. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-93-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-93-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span> is invertible, because its columns are assumed to be independent.<\/span><\/p><h3 id='2-remarks'><span>(2) Remarks<\/span><\/h3><ul><li><span>If a matrix has no repeated eigenvalues, then its eigenvectors are automatically independent.<\/span><\/li><\/ul><p><em><span>Any matrix with distinct eigenvalues can be diagonalized.<\/span><\/em><\/p><ul><li><span>The diagonalizing matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-93-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-93-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span> is not unique.<\/span><\/li><li><span>The order of the eigenvectors in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-93-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-93-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><span> and the eigenvalues in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.57ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 694 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-94-TEX-N-39B\" d=\"M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"39B\" xlink:href=\"#MJX-94-TEX-N-39B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u039b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Lambda<\/script><span> is automatically the same.<\/span><\/li><li><span>Not all matrices possess n linearly independent eigenvectors.<\/span><\/li><\/ul><p><em><span>Not all matrices are diagonalizable.<\/span><\/em><\/p><blockquote><p><span>Diaginalizability of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> depends on enough eigenvectors. (n independent eigenvectors)<\/span><\/p><p><span>Invertibility of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> depends on nonzero eigenvalues. (no zero eigenvalues)<\/span><\/p><\/blockquote><ul><li><span>There is no connection between diagonalizability and invertibility.<\/span><\/li><li><span>Diagonalization can fail only if there are repeated eigenvalues.<\/span><\/li><\/ul><p><span>But it does not always fail. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.854ex\" height=\"1.805ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 2587.6 798\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-97-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-97-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-97-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-97-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1027.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-97-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2083.6,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-97-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>=<\/mo><mi>I<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A=I<\/script><span> has repeated eigenvalues <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.43ex\" height=\"1.946ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 4168 860\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-98-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-98-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-98-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-98-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(500,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-98-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(944.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-98-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1444.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-98-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1889.3,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-98-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2334,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-98-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2778.7,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-98-TEX-N-2E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3223.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-98-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3668,0)\"><use data-c=\"31\" xlink:href=\"#MJX-98-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo>,<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>,<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1, 1, ..., 1<\/script><span> but it is already diagonal.<\/span><\/p><h2 id='3-power-and-products'><span>3. Power and Products<\/span><\/h2><ul><li><span>The eigenvalues of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.718ex\" height=\"1.932ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 1201.4 853.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-99-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-99-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-99-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(783,363) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-99-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>A<\/mi><mi>k<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A^k<\/script><span> has the same eigenvectors as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span>, and eigenvalues <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.183ex\" height=\"2.601ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -853.7 5827.1 1149.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.669ex;\"><defs><path id=\"MJX-101-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><path id=\"MJX-101-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-101-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-101-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-101-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-101-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-101-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 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transform=\"translate(4736.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-101-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,363) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-101-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-247) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-101-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>\u03bb<\/mi><mn>1<\/mn><mi>k<\/mi><\/msubsup><mo>,<\/mo><msubsup><mi>\u03bb<\/mi><mn>2<\/mn><mi>k<\/mi><\/msubsup><mo>,<\/mo><mo>.<\/mo><mo>.<\/mo><mo>.<\/mo><mo>,<\/mo><msubsup><mi>\u03bb<\/mi><mi>n<\/mi><mi>k<\/mi><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda_1^k, \\lambda_2^k,..., \\lambda_n^k<\/script><span>.<\/span><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" 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data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msup><mi>A<\/mi><mn>2<\/mn><\/msup><mi>x<\/mi><mo>=<\/mo><mi>A<\/mi><mi>\u03bb<\/mi><mi>x<\/mi><mo>=<\/mo><mi>\u03bb<\/mi><mi>A<\/mi><mi>x<\/mi><mo>=<\/mo><msup><mi>\u03bb<\/mi><mn>2<\/mn><\/msup><mi>x<\/mi><mo>.<\/mo><mtext>&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mi>c<\/mi><mi>a<\/mi><mi>r<\/mi><mi>e<\/mi><mi>f<\/mi><mi>u<\/mi><mi>l<\/mi><mo>!<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-102-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-102-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A<\/script><span> is invertible, the eigenvalues of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.929ex\" height=\"1.887ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 1736.7 833.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-103-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 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data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-103-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-103-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>A<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">A^{-1}<\/script><span> are <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.321ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1910 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-104-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 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83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-104-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-104-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(1000,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-104-TEX-I-1D706\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(616,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-104-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>\u03bb<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1\/\\lambda_i<\/script><span>.<\/span><\/li><\/ul><h1 id='chapter-6-positive-definite-matrices'><span>Chapter 6 Positive Definite Matrices<\/span><\/h1><h2 id='1-minima-maxima-and-saddle-points'><span>1. Minima, Maxima and Saddle Points<\/span><\/h2><p>&nbsp;<\/p><\/div><\/div>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Chapter 4 Determinants &nbsp; Chapter 4 Determinants 1.<a class=\"more-link\" href=\"https:\/\/blog.mhrooz.xyz\/index.php\/2022\/12\/26\/recap_linear_algebra\/\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">&#8220;Recap: Linear Algebra&#8221;<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[49],"tags":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2488"}],"collection":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=2488"}],"version-history":[{"count":6,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2488\/revisions"}],"predecessor-version":[{"id":2513,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2488\/revisions\/2513"}],"wp:attachment":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=2488"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=2488"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=2488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}