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hilbert_min_max.html
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<!doctype html>
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<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
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</style><title>hilbert_min_max</title>
</head>
<body class='typora-export'><div class='typora-export-content'>
<div id='write' class=''><h1 id='hilberts-min-max-theorem'><span>Hilbert's Min-Max Theorem</span></h1><ul><li><strong><span>Fact-1</span></strong><span>: If </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.88ex" height="1.602ex" role="img" focusable="false" viewBox="0 -697 831 708" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1655-TEX-B-1D402" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D402" xlink:href="#MJX-1655-TEX-B-1D402"></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="ORD"><mi mathvariant="bold">C</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{C}</script><span> is a real symmetric matrix, then it has an orthonormal basis of eigenvectors with real eigenvalues.</span></li><li><strong><span>Fact-2</span></strong><span>: If </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.88ex" height="1.602ex" role="img" focusable="false" viewBox="0 -697 831 708" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1655-TEX-B-1D402" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D402" xlink:href="#MJX-1655-TEX-B-1D402"></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="ORD"><mi mathvariant="bold">C</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{C}</script><span> is a symmetric, positive semi-definite matrix, the eigenvalues are non-negative.</span></li><li><strong><span>Fact-3</span></strong><span>: The covariance matrix is symmetric and positive semi-definite.</span></li></ul><p><span>From the these three facts, we can conclude that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.88ex" height="1.602ex" role="img" focusable="false" viewBox="0 -697 831 708" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1655-TEX-B-1D402" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D402" xlink:href="#MJX-1655-TEX-B-1D402"></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="ORD"><mi mathvariant="bold">C</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{C}</script><span> has a basis of eigenvectors </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.071ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5777.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 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mathvariant="bold">w</mi></mrow><mi>d</mi></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\{\mathbf{w}_1, \cdots, \mathbf{w}_d\}</script><span> with eigenvalues </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="17.479ex" height="1.926ex" role="img" focusable="false" viewBox="0 -694 7725.9 851.1" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.355ex;"><defs><path id="MJX-1657-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-1657-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 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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-1657-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="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1657-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1657-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1297.3,0)"><use data-c="2265" xlink:href="#MJX-1657-TEX-N-2265"></use></g><g data-mml-node="mo" transform="translate(2353.1,0)"><use data-c="22EF" xlink:href="#MJX-1657-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(3802.9,0)"><use data-c="2265" xlink:href="#MJX-1657-TEX-N-2265"></use></g><g data-mml-node="msub" transform="translate(4858.7,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1657-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D451" xlink:href="#MJX-1657-TEX-I-1D451"></use></g></g><g data-mml-node="mo" transform="translate(6170.1,0)"><use data-c="2265" xlink:href="#MJX-1657-TEX-N-2265"></use></g><g data-mml-node="mn" transform="translate(7225.9,0)"><use data-c="30" xlink:href="#MJX-1657-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mi>λ</mi><mi>d</mi></msub><mo>≥</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_1 \geq \cdots \geq \lambda_d \geq 0</script><span>. Now, we are interested in the following optimization problem:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n10" cid="n10" 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" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="16.36ex" height="4.245ex" role="img" focusable="false" viewBox="0 -891.7 7231.2 1876.1" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.227ex;"><defs><path id="MJX-1641-TEX-N-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-1641-TEX-N-61" d="M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z"></path><path id="MJX-1641-TEX-N-78" d="M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 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data-mjx-texclass="ORD" transform="translate(6400.2,0)"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1641-TEX-B-1D430"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munder><mi>max</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mo>,</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo>=</mo><mn>1</mn></mrow></munder><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>T</mi></msup><mi>C</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>Consider any arbitrary vector </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.446ex" height="1.032ex" role="img" focusable="false" viewBox="0 -450 639 456" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.014ex;"><defs><path id="MJX-1658-TEX-B-1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D42E" xlink:href="#MJX-1658-TEX-B-1D42E"></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="ORD"><mi mathvariant="bold">u</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{u}</script><span> that can be uniquely expressed in terms of the orthonormal basis:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n12" cid="n12" 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" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.914ex" height="6.757ex" role="img" focusable="false" viewBox="0 -1740.7 5708.1 2986.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-1642-TEX-B-1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z"></path><path id="MJX-1642-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-1642-TEX-LO-2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 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stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D42E" xlink:href="#MJX-1642-TEX-B-1D42E"></use></g></g><g data-mml-node="mo" transform="translate(916.8,0)"><use data-c="3D" xlink:href="#MJX-1642-TEX-N-3D"></use></g><g data-mml-node="munderover" transform="translate(1972.6,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-1642-TEX-LO-2211"></use></g><g data-mml-node="TeXAtom" transform="translate(148.2,-1087.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-1642-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-1642-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-1642-TEX-N-31"></use></g></g><g data-mml-node="TeXAtom" transform="translate(538.2,1150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-1642-TEX-I-1D451"></use></g></g></g><g data-mml-node="msub" transform="translate(3583.2,0)"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-1642-TEX-I-1D6FC"></use></g><g data-mml-node="mi" transform="translate(673,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1642-TEX-I-1D456"></use></g></g><g data-mml-node="msub" transform="translate(4550.2,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1642-TEX-B-1D430"></use></g></g><g data-mml-node="mi" transform="translate(864,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1642-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo>=</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container></div></div><p><span>Since we are only looking for vectors with unit norm, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.11ex" height="2.26ex" role="img" focusable="false" viewBox="0 -749.5 3584.6 999" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.564ex;"><defs><path id="MJX-1659-TEX-N-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-1659-TEX-B-1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z"></path><path id="MJX-1659-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-1659-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="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-1659-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(278,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-1659-TEX-N-7C"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(556,0)"><g data-mml-node="mi"><use data-c="1D42E" xlink:href="#MJX-1659-TEX-B-1D42E"></use></g></g><g data-mml-node="mo" transform="translate(1195,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-1659-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1473,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-1659-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(2028.8,0)"><use data-c="3D" xlink:href="#MJX-1659-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3084.6,0)"><use data-c="31" xlink:href="#MJX-1659-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">||\mathbf{u}|| = 1</script><span>. This implies:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n14" cid="n14" 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" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="24.802ex" height="6.354ex" role="img" focusable="false" viewBox="0 -1562.5 10962.7 2808.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-1643-TEX-B-1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 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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-1643-TEX-N-27F9" d="M1218 514Q1218 525 1234 525Q1239 525 1242 525T1247 525T1251 524T1253 523T1255 520T1257 517T1260 512Q1297 438 1358 381T1469 300T1565 263Q1582 258 1582 250T1573 239T1536 228T1478 204Q1334 134 1260 -12Q1256 -21 1253 -22T1238 -24Q1218 -24 1218 -17Q1218 -13 1223 0Q1258 69 1309 123L1319 133H70Q56 140 56 153Q56 168 72 173H1363L1373 181Q1412 211 1490 250Q1489 251 1472 259T1427 283T1373 319L1363 327H710L707 328L390 327H72Q56 332 56 347Q56 360 70 367H1319L1309 377Q1276 412 1247 458T1218 514Z"></path><path id="MJX-1643-TEX-LO-2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 252Q725 250 724 246Q721 243 460 -56L196 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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-1643-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 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data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-1643-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-1643-TEX-N-31"></use></g></g><g data-mml-node="TeXAtom" transform="translate(509.9,1150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1643-TEX-I-1D45B"></use></g></g></g><g data-mml-node="msubsup" transform="translate(8052.6,0)"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-1643-TEX-I-1D6FC"></use></g><g data-mml-node="mn" transform="translate(673,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-1643-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(673,-247) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1643-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(9406.9,0)"><use data-c="3D" xlink:href="#MJX-1643-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(10462.7,0)"><use data-c="31" xlink:href="#MJX-1643-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mi>T</mi></msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo>=</mo><mn>1</mn><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo stretchy="false">⟹</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></munderover><msubsup><mi>α</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></div></div><p><span>The above result is true because of the orthonormality of the basis vectors. 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transform="translate(0,-5773.5)"><g data-mml-node="mtd" transform="translate(2689.8,0)"></g><g data-mml-node="mtd" transform="translate(2689.8,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="2264" xlink:href="#MJX-1644-TEX-N-2264"></use></g><g data-mml-node="msub" transform="translate(1333.6,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1644-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1644-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(2575.3,0)"><use data-c="22C5" xlink:href="#MJX-1644-TEX-N-22C5"></use></g><g data-mml-node="mrow" transform="translate(3075.6,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="28" xlink:href="#MJX-1644-TEX-S4-28"></use></g><g data-mml-node="munderover" transform="translate(792,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-1644-TEX-LO-2211"></use></g><g 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xlink:href="#MJX-1644-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(3479.2,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJX-1644-TEX-S4-29"></use></g></g></g></g><g data-mml-node="mtr" transform="translate(0,-8073)"><g data-mml-node="mtd" transform="translate(2689.8,0)"></g></g><g data-mml-node="mtr" transform="translate(0,-9373)"><g data-mml-node="mtd" transform="translate(2689.8,0)"></g><g data-mml-node="mtd" transform="translate(2689.8,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-1644-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(1333.6,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1644-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1644-TEX-N-31"></use></g></g></g></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" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mi>T</mi></msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow></mtd><mtd><mi></mi><mo>=</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>i</mi></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>T</mi></msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></munderover><msub><mi>α</mi><mi>j</mi></msub><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>j</mi></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></munderover><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mi>α</mi><mi>j</mi></msub><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>i</mi><mi>T</mi></msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>j</mi></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></munderover><msubsup><mi>α</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>λ</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>≤</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>⋅</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></munderover><msubsup><mi>α</mi><mi>i</mi><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>λ</mi><mn>1</mn></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>We see that the objective function </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.086ex" height="1.929ex" role="img" focusable="false" viewBox="0 -841.7 2689.8 852.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1661-TEX-B-1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z"></path><path id="MJX-1661-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><path id="MJX-1661-TEX-B-1D402" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D42E" xlink:href="#MJX-1661-TEX-B-1D42E"></use></g></g><g data-mml-node="mi" transform="translate(672,363) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-1661-TEX-I-1D447"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1219.8,0)"><g data-mml-node="mi"><use data-c="1D402" xlink:href="#MJX-1661-TEX-B-1D402"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2050.8,0)"><g data-mml-node="mi"><use data-c="1D42E" xlink:href="#MJX-1661-TEX-B-1D42E"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mi>T</mi></msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{u}^T \mathbf{C} \mathbf{u}</script><span> is bounded above by </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.307ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 1019.6 844" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1666-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-1666-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="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1666-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1666-TEX-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_1</script><span>. This value of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.307ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 1019.6 844" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1666-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-1666-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="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1666-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1666-TEX-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_1</script><span> occurs when </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.331ex" height="1.658ex" role="img" focusable="false" viewBox="0 -583 3240.1 733" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1664-TEX-B-1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z"></path><path id="MJX-1664-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-1664-TEX-B-1D430" d="M624 444Q636 441 722 441Q797 441 800 444H805V382H741L593 11Q592 10 590 8T586 4T584 2T581 0T579 -2T575 -3T571 -3T567 -4T561 -4T553 -4H542Q525 -4 518 6T490 70Q474 110 463 137L415 257L367 137Q357 111 341 72Q320 17 313 7T289 -4H277Q259 -4 253 -2T238 11L90 382H25V444H32Q47 441 140 441Q243 441 261 444H270V382H222L310 164L382 342L366 382H303V444H310Q322 441 407 441Q508 441 523 444H531V382H506Q481 382 481 380Q482 376 529 259T577 142L674 382H617V444H624Z"></path><path id="MJX-1664-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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D42E" xlink:href="#MJX-1664-TEX-B-1D42E"></use></g></g><g data-mml-node="mo" transform="translate(916.8,0)"><use data-c="3D" xlink:href="#MJX-1664-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(1972.6,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1664-TEX-B-1D430"></use></g></g><g data-mml-node="mn" transform="translate(864,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1664-TEX-N-31"></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="ORD"><mi mathvariant="bold">u</mi></mrow><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{u} = \mathbf{w}_1</script><span>. This is one way of proving the theorem. In the steps given above, we have used the fact that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.196ex" height="1.934ex" role="img" focusable="false" viewBox="0 -697 5390.4 854.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-1665-TEX-B-1D402" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path><path id="MJX-1665-TEX-B-1D430" d="M624 444Q636 441 722 441Q797 441 800 444H805V382H741L593 11Q592 10 590 8T586 4T584 2T581 0T579 -2T575 -3T571 -3T567 -4T561 -4T553 -4H542Q525 -4 518 6T490 70Q474 110 463 137L415 257L367 137Q357 111 341 72Q320 17 313 7T289 -4H277Q259 -4 253 -2T238 11L90 382H25V444H32Q47 441 140 441Q243 441 261 444H270V382H222L310 164L382 342L366 382H303V444H310Q322 441 407 441Q508 441 523 444H531V382H506Q481 382 481 380Q482 376 529 259T577 142L674 382H617V444H624Z"></path><path id="MJX-1665-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 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><path id="MJX-1665-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-1665-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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D402" xlink:href="#MJX-1665-TEX-B-1D402"></use></g></g><g data-mml-node="msub" transform="translate(831,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1665-TEX-B-1D430"></use></g></g><g data-mml-node="mi" transform="translate(864,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1665-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(2266.7,0)"><use data-c="3D" xlink:href="#MJX-1665-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(3322.5,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1665-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1665-TEX-I-1D456"></use></g></g><g data-mml-node="msub" transform="translate(4232.5,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1665-TEX-B-1D430"></use></g></g><g data-mml-node="mi" transform="translate(864,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1665-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"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>i</mi></msub><mo>=</mo><msub><mi>λ</mi><mi>i</mi></msub><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{C} \mathbf{w}_i = \lambda_i \mathbf{w}_i</script><span>, the orthonormality of the eigenvectors and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.307ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 1019.6 844" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1666-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-1666-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="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1666-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1666-TEX-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_1</script><span> being the largest eigenvalue.</span></p></div></div>
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