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variance.html
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variance.html
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</style><title>variance</title>
</head>
<body class='typora-export'><div class='typora-export-content'>
<div id='write' class=''><h1 id='pca-as-variance-maximization'><span>PCA as Variance Maximization</span></h1><p><span>This document addresses the following questions:</span></p><ul><li><span>What do we mean by variance in a dataset?</span></li><li><span>Why is variance important?</span></li><li><span>Why do we center a dataset before doing PCA?</span></li><li><span>What are the principal components?</span></li></ul><p><strong><span>Note</span></strong><span>: While reading through this post please make a note of the difference in terminology:</span></p><ul><li><span>total variance of a dataset</span></li><li><span>variance along a particular direction</span></li></ul><p> </p><h2 id='total-variance'><span>Total Variance</span></h2><p><span>Consider </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" 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">n</script><span> data-points in </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.653ex" height="1.932ex" role="img" focusable="false" viewBox="0 -853.7 1172.7 853.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-1627-TEX-D-211D" d="M17 665Q17 672 28 683H221Q415 681 439 677Q461 673 481 667T516 654T544 639T566 623T584 607T597 592T607 578T614 565T618 554L621 548Q626 530 626 497Q626 447 613 419Q578 348 473 326L455 321Q462 310 473 292T517 226T578 141T637 72T686 35Q705 30 705 16Q705 7 693 -1H510Q503 6 404 159L306 310H268V183Q270 67 271 59Q274 42 291 38Q295 37 319 35Q344 35 353 28Q362 17 353 3L346 -1H28Q16 5 16 16Q16 35 55 35Q96 38 101 52Q106 60 106 341T101 632Q95 645 55 648Q17 648 17 665ZM241 35Q238 42 237 45T235 78T233 163T233 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xlink:href="#MJX-1577-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(4885.8,0)"><use data-c="7D" xlink:href="#MJX-1577-TEX-N-7D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo fence="false" stretchy="false">{</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>n</mi></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>The mean is given by:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n22" cid="n22" 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="13.106ex" height="6.484ex" role="img" focusable="false" viewBox="0 -1620 5792.8 2865.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-1578-TEX-BI-1D741" d="M294 -8Q265 -8 244 -5T213 1T201 4Q200 4 192 -32T172 -111T155 -168Q134 -211 86 -211Q62 -211 48 -196T34 -158Q37 -144 103 123T174 404Q182 424 201 438T244 452Q271 452 284 436T298 404Q298 392 267 269T235 114Q235 43 305 43Q342 43 375 68T418 110Q420 112 455 253T492 397Q514 444 562 444Q587 444 601 429T615 397Q615 387 599 320T563 178T542 93Q540 81 540 72Q540 42 558 42Q580 42 596 75Q606 94 616 134Q621 155 624 158T646 162H651H662Q682 162 682 148Q681 142 679 132T665 94T641 47T602 9T548 -8Q523 -8 502 -3T468 11T446 27T432 40L429 46Q367 -8 294 -8Z"></path><path id="MJX-1578-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-1578-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-1578-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-1578-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 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 924Q55 945 60 948Z"></path><path id="MJX-1578-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-1578-TEX-B-1D431" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></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="1D741" xlink:href="#MJX-1578-TEX-BI-1D741"></use></g><g data-mml-node="mo" transform="translate(985.8,0)"><use data-c="3D" xlink:href="#MJX-1578-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2041.6,0)"><g data-mml-node="mrow" transform="translate(270,760)"><g data-mml-node="mpadded"><g data-mml-node="mrow"></g></g><g data-mml-node="mstyle"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-1578-TEX-N-31"></use></g></g></g></g><g data-mml-node="mrow" transform="translate(220,-820)"><g data-mml-node="mpadded"><g data-mml-node="mrow"></g></g><g data-mml-node="mstyle"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1578-TEX-I-1D45B"></use></g></g></g></g><rect width="800" height="60" x="120" y="220"></rect></g><g data-mml-node="munderover" transform="translate(3248.2,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-1578-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-1578-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-1578-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-1578-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-1578-TEX-I-1D45B"></use></g></g></g><g data-mml-node="msub" transform="translate(4858.9,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1578-TEX-B-1D431"></use></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1578-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"><mi mathvariant="bold-italic">μ</mi><mo>=</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></mstyle></mrow></mfrac><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><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container></div></div><p><span>The covariance matrix is given by:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n24" cid="n24" 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="29.864ex" height="6.484ex" role="img" focusable="false" viewBox="0 -1620 13199.8 2865.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-1579-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-1579-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-1579-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-1579-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-1579-TEX-N-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-1579-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 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 924Q55 945 60 948Z"></path><path id="MJX-1579-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-1579-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-1579-TEX-B-1D431" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path><path id="MJX-1579-TEX-B-1D422" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path><path id="MJX-1579-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-1579-TEX-BI-1D741" d="M294 -8Q265 -8 244 -5T213 1T201 4Q200 4 192 -32T172 -111T155 -168Q134 -211 86 -211Q62 -211 48 -196T34 -158Q37 -144 103 123T174 404Q182 424 201 438T244 452Q271 452 284 436T298 404Q298 392 267 269T235 114Q235 43 305 43Q342 43 375 68T418 110Q420 112 455 253T492 397Q514 444 562 444Q587 444 601 429T615 397Q615 387 599 320T563 178T542 93Q540 81 540 72Q540 42 558 42Q580 42 596 75Q606 94 616 134Q621 155 624 158T646 162H651H662Q682 162 682 148Q681 142 679 132T665 94T641 47T602 9T548 -8Q523 -8 502 -3T468 11T446 27T432 40L429 46Q367 -8 294 -8Z"></path><path id="MJX-1579-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-1579-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(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-1579-TEX-B-1D402"></use></g></g><g data-mml-node="mo" transform="translate(1108.8,0)"><use data-c="3D" xlink:href="#MJX-1579-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2164.6,0)"><g 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transform="translate(6897.8,0)"><use data-c="2212" xlink:href="#MJX-1579-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(7898,0)"><use data-c="1D741" xlink:href="#MJX-1579-TEX-BI-1D741"></use></g><g data-mml-node="mo" transform="translate(8606,0)"><use data-c="29" xlink:href="#MJX-1579-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(8995,0)"><use data-c="28" xlink:href="#MJX-1579-TEX-N-28"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(9384,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1579-TEX-B-1D431"></use></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D422" xlink:href="#MJX-1579-TEX-B-1D422"></use></g></g></g><g data-mml-node="mo" transform="translate(10521.8,0)"><use data-c="2212" xlink:href="#MJX-1579-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(11522,0)"><use data-c="1D741" xlink:href="#MJX-1579-TEX-BI-1D741"></use></g><g data-mml-node="msup" transform="translate(12230,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-1579-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(422,413) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-1579-TEX-I-1D447"></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">C</mi></mrow><mo>=</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></mstyle></mrow></mfrac><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>n</mi></mrow></munderover><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">i</mi></msub></mrow><mo>−</mo><mi mathvariant="bold-italic">μ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">i</mi></msub></mrow><mo>−</mo><mi mathvariant="bold-italic">μ</mi><msup><mo stretchy="false">)</mo><mi>T</mi></msup></math></mjx-assistive-mml></mjx-container></div></div><p><span>The variance of the dataset along the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.866ex" height="1.956ex" role="img" focusable="false" viewBox="0 -853.7 1266.6 864.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1634-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-1634-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-1634-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></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="1D458" xlink:href="#MJX-1634-TEX-I-1D458"></use></g><g data-mml-node="TeXAtom" transform="translate(554,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-1634-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(361,0)"><use data-c="210E" xlink:href="#MJX-1634-TEX-I-210E"></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>k</mi><mrow data-mjx-texclass="ORD"><mi>t</mi><mi>h</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">k^{th}</script><span> direction in the standard basis is </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.472ex" height="1.952ex" role="img" focusable="false" viewBox="0 -705 1534.8 862.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-1615-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path id="MJX-1615-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="msub"><g data-mml-node="mi"><use data-c="1D436" xlink:href="#MJX-1615-TEX-I-1D436"></use></g><g data-mml-node="TeXAtom" transform="translate(748,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-1615-TEX-I-1D458"></use></g><g data-mml-node="mi" transform="translate(521,0)"><use data-c="1D458" xlink:href="#MJX-1615-TEX-I-1D458"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>k</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">C_{kk}</script><span>, the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.866ex" height="1.956ex" role="img" focusable="false" viewBox="0 -853.7 1266.6 864.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1634-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-1634-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-1634-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></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="1D458" xlink:href="#MJX-1634-TEX-I-1D458"></use></g><g data-mml-node="TeXAtom" transform="translate(554,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-1634-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(361,0)"><use data-c="210E" xlink:href="#MJX-1634-TEX-I-210E"></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>k</mi><mrow data-mjx-texclass="ORD"><mi>t</mi><mi>h</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">k^{th}</script><span> element in the diagonal of </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-1633-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-1633-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>. 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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-1580-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-1580-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 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data-mml-node="mstyle"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1580-TEX-I-1D45B"></use></g></g></g></g><rect width="800" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(6486.5,0)"><use data-c="22C5" xlink:href="#MJX-1580-TEX-N-22C5"></use></g><g data-mml-node="munderover" transform="translate(6986.8,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-1580-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-1580-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-1580-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-1580-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-1580-TEX-I-1D45B"></use></g></g></g><g data-mml-node="mo" transform="translate(8430.8,0)"><use data-c="28" xlink:href="#MJX-1580-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(8819.8,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-1580-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-1580-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D458" xlink:href="#MJX-1580-TEX-I-1D458"></use></g></g></g><g data-mml-node="mo" transform="translate(10309.3,0)"><use data-c="2212" xlink:href="#MJX-1580-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(11309.6,0)"><g data-mml-node="mi"><use data-c="1D707" xlink:href="#MJX-1580-TEX-I-1D707"></use></g><g data-mml-node="mi" transform="translate(636,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-1580-TEX-I-1D458"></use></g></g><g data-mml-node="msup" transform="translate(12364,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-1580-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-1580-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>σ</mi><mi>k</mi><mn>2</mn></msubsup><mo>=</mo><msub><mi>C</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></mstyle></mrow></mfrac><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>n</mi></mrow></munderover><mo stretchy="false">(</mo><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></msub><mo>−</mo><msub><mi>μ</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container></div></div><p><span>Here, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.867ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 1267.4 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-1618-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 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664Q129 683 137 683Z"></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="1D458" xlink:href="#MJX-1634-TEX-I-1D458"></use></g><g data-mml-node="TeXAtom" transform="translate(554,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-1634-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(361,0)"><use data-c="210E" xlink:href="#MJX-1634-TEX-I-210E"></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>k</mi><mrow data-mjx-texclass="ORD"><mi>t</mi><mi>h</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">k^{th}</script><span> feature in the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.467ex" 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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-1620-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></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 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One way of defining the </span><strong><span>total variance</span></strong><span> of the dataset is as follows:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n28" cid="n28" 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="45.589ex" height="6.802ex" role="img" focusable="false" viewBox="0 -1740.7 20150.2 3006.4" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.864ex;"><defs><path id="MJX-1581-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 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mean. Since this quantity involves distances, it is independent of the basis we choose to represent the data-points. </span></p><p><span>Why are we interested in the idea of variance? The total variance is a measure of the information contained in the dataset. Imagine a dataset with zero variance. This would correspond to the situation where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.568ex" height="1.676ex" role="img" focusable="false" viewBox="0 -583 5996.9 740.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-1621-TEX-B-1D431" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path><path id="MJX-1621-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-1621-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-1621-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-1621-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="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1621-TEX-B-1D431"></use></g></g><g data-mml-node="mn" transform="translate(640,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1621-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1321.3,0)"><use data-c="3D" xlink:href="#MJX-1621-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2377.1,0)"><use data-c="22EF" xlink:href="#MJX-1621-TEX-N-22EF"></use></g><g data-mml-node="mo" transform="translate(3826.9,0)"><use data-c="3D" xlink:href="#MJX-1621-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(4882.7,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1621-TEX-B-1D431"></use></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-1621-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"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mn>1</mn></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{x}_1 = \cdots = \mathbf{x}_n</script><span>. There is nothing to analyze here as the dataset is just a single observation repeated </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-1622-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-1622-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> times. Variability is therefore seen as a measure of the information present in the data. The next conceptual step from here is to look for those directions which capture most of this information. The motivation for this is to get as succinct a representation as possible using the fewest directions. This is the theme of compression as a route to comprehension which permeates all unsupervised learning approaches we have seen so far. Now, let us take a short detour and understand the idea of centering.</span></p><p> </p><h2 id='centering'><span>Centering</span></h2><p><span>A clear computational advantage of centering the dataset is that it makes all the expressions less cluttered:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n34" cid="n34" 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="45.41ex" height="6.484ex" role="img" focusable="false" viewBox="0 -1620 20071.3 2865.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-1582-TEX-B-1D402" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 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data-mml-node="mo" transform="translate(19078.7,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-1582-TEX-N-7C"></use></g><g data-mml-node="msup" transform="translate(19356.7,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-1582-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(311,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-1582-TEX-N-32"></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">C</mi></mrow><mo>=</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></mstyle></mrow></mfrac><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>n</mi></mrow></munderover><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi></msub><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi><mi>T</mi></msubsup><mo>,</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mtext>trace</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></mstyle></mrow><mrow><mpadded height="8.6pt" depth="3pt" width="0"><mrow></mrow></mpadded><mstyle displaystyle="false" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></mstyle></mrow></mfrac><mo>⋅</mo><mo data-mjx-texclass="OP">∑</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi></msub><mo data-mjx-texclass="ORD" stretchy="false">|</mo><msup><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container></div></div><p><span>A more principled argument is to note that we are looking at lines (directions) that pass through the origin. Why are we looking for such lines? 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374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></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="1D43E" xlink:href="#MJX-1623-TEX-I-1D43E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">K</script><span>-dimensional linear subspace of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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We are trying to find the basis elements (directions) of this linear subspace. Since we are dealing with vector spaces, the zero-vector or the origin plays an important role here. </span></p><p><span>For a more practical understanding of this concept, consider the data collected from a weighing machine. Because of a fault in the machine's display, all readings are five more than the true readings. In other words, there is a constant offset of five units. By centering the dataset, we also end up removing this offset. Extending this idea to higher dimensions, the important directions in the non-centered dataset will be influenced by this offset.</span></p><p><span>We will now continue with the centered dataset from this stage.</span></p><p> </p><h2 id='principal-components'><span>Principal Components</span></h2><p><span>If </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.057ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5329.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-1626-TEX-N-7B" d="M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 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transform="translate(4829.2,0)"><use data-c="7D" xlink:href="#MJX-1626-TEX-N-7D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mi>d</mi></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\{\mathbf{v}_1, \cdots, \mathbf{v}_d\}</script><span> is an orthonormal basis of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.653ex" height="1.932ex" role="img" focusable="false" viewBox="0 -853.7 1172.7 853.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-1627-TEX-D-211D" d="M17 665Q17 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464 243Z"></path><path id="MJX-1627-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></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="211D" xlink:href="#MJX-1627-TEX-D-211D"></use></g></g><g data-mml-node="TeXAtom" transform="translate(755,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-1627-TEX-I-1D451"></use></g></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="double-struck">R</mi></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbb{R}^{d}</script><span>, then for each point </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.113ex" height="1.361ex" role="img" focusable="false" viewBox="0 -444 934 601.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-1628-TEX-B-1D431" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path><path id="MJX-1628-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></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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1628-TEX-B-1D431"></use></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1628-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"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{x}_i</script><span> we have:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n41" cid="n41" 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="17.285ex" height="6.952ex" role="img" focusable="false" viewBox="0 -1740.7 7640 3072.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.014ex;"><defs><path id="MJX-1583-TEX-B-1D431" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path><path id="MJX-1583-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-1583-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-1583-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 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 -332T1204 -283T1307 -196Q1328 -170 1348 -124H1388Q1388 -125 1381 -145T1356 -210T1325 -294L1267 -449L666 -450Q64 -450 61 -448Q55 -446 55 -439Q55 -437 57 -433L590 177Q590 178 557 222T452 366T322 544L56 909L55 924Q55 945 60 948Z"></path><path id="MJX-1583-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path id="MJX-1583-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-1583-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-1583-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-1583-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-1583-TEX-B-1D42F" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path id="MJX-1583-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></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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1583-TEX-B-1D431"></use></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1583-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1211.7,0)"><use data-c="3D" xlink:href="#MJX-1583-TEX-N-3D"></use></g><g data-mml-node="munderover" transform="translate(2267.5,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-1583-TEX-LO-2211"></use></g><g data-mml-node="TeXAtom" transform="translate(124.5,-1087.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D457" xlink:href="#MJX-1583-TEX-I-1D457"></use></g><g data-mml-node="mo" transform="translate(412,0)"><use data-c="3D" xlink:href="#MJX-1583-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1190,0)"><use data-c="31" xlink:href="#MJX-1583-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-1583-TEX-I-1D451"></use></g></g></g><g data-mml-node="mo" transform="translate(3711.5,0)"><use data-c="28" xlink:href="#MJX-1583-TEX-N-28"></use></g><g data-mml-node="msubsup" transform="translate(4100.5,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1583-TEX-B-1D431"></use></g></g><g data-mml-node="mi" transform="translate(640,413) scale(0.707)"><use data-c="1D447" xlink:href="#MJX-1583-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(640,-247) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1583-TEX-I-1D456"></use></g></g><g data-mml-node="msub" transform="translate(5288.3,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D42F" xlink:href="#MJX-1583-TEX-B-1D42F"></use></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-1583-TEX-I-1D457"></use></g></g><g data-mml-node="mo" transform="translate(6269.6,0)"><use data-c="29" xlink:href="#MJX-1583-TEX-N-29"></use></g><g data-mml-node="msub" transform="translate(6658.6,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D42F" xlink:href="#MJX-1583-TEX-B-1D42F"></use></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-1583-TEX-I-1D457"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi></msub><mo>=</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><mo stretchy="false">(</mo><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi><mi>T</mi></msubsup><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mi>j</mi></msub><mo stretchy="false">)</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container></div></div><p><span>The variance along the direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.395ex" height="1.361ex" role="img" focusable="false" viewBox="0 -444 1058.4 601.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-1629-TEX-B-1D42F" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path id="MJX-1629-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="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D42F" xlink:href="#MJX-1629-TEX-B-1D42F"></use></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-1629-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"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{v}_k</script><span> is:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n43" cid="n43" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div 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Note the change in terminology. We are not worried about the total variance any longer, as that is a constant for a given dataset. We are now interested in </span><strong><span>variances along particular directions</span></strong><span>. The most important direction happens to be </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.868ex" height="1.344ex" role="img" focusable="false" viewBox="0 -444 1267.6 594" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1630-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-1630-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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1630-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-1630-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><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{w}_1</script><span>, the eigenvector corresponding to the largest eigenvalue, </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-1632-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-1632-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-1632-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1632-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>, a consequence of the Hilbert Min-Max theorem. 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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="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1635-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-1635-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"><msub><mi>λ</mi><mi>k</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_k</script><span> with the important property that:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n47" cid="n47" 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="23.225ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 10265.7 851.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-1586-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-1586-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-1586-TEX-N-2265" d="M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path id="MJX-1586-TEX-N-22EF" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250ZM525 250Q525 274 542 292T585 310Q609 310 627 294T646 251Q646 226 629 208T586 190T543 207T525 250ZM972 250Q972 274 989 292T1032 310Q1056 310 1074 294T1093 251Q1093 226 1076 208T1033 190T990 207T972 250Z"></path><path id="MJX-1586-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-1586-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-1586-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-1586-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1586-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1297.3,0)"><use data-c="2265" xlink:href="#MJX-1586-TEX-N-2265"></use></g><g data-mml-node="mo" transform="translate(2353.1,0)"><use data-c="22EF" xlink:href="#MJX-1586-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(3691.8,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1586-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-1586-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(5004,0)"><use data-c="2265" xlink:href="#MJX-1586-TEX-N-2265"></use></g><g data-mml-node="mo" transform="translate(6059.7,0)"><use data-c="22EF" xlink:href="#MJX-1586-TEX-N-22EF"></use></g><g data-mml-node="msub" transform="translate(7398.4,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1586-TEX-I-1D706"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D451" xlink:href="#MJX-1586-TEX-I-1D451"></use></g></g><g data-mml-node="mo" transform="translate(8709.9,0)"><use data-c="2265" xlink:href="#MJX-1586-TEX-N-2265"></use></g><g data-mml-node="mn" transform="translate(9765.7,0)"><use data-c="30" xlink:href="#MJX-1586-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>λ</mi><mn>1</mn></msub><mo>≥</mo><mo>⋯</mo><msub><mi>λ</mi><mi>k</mi></msub><mo>≥</mo><mo>⋯</mo><msub><mi>λ</mi><mi>d</mi></msub><mo>≥</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></div></div><p><span>Because of this property, we say that the top-</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1637-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="mi"><use data-c="1D458" xlink:href="#MJX-1637-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span> directions capture the most important information contained in the dataset and we typically look for that value of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1637-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="mi"><use data-c="1D458" xlink:href="#MJX-1637-TEX-I-1D458"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">k</script><span> which captures </span><mjx-container class="MathJax" jax="SVG" style="position: 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The total variance of the dataset remains unchanged as expected:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n49" cid="n49" 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="24.393ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10781.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-1587-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 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transform="translate(10392.7,0)"><use data-c="29" xlink:href="#MJX-1587-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>λ</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>λ</mi><mi>d</mi></msub><mo>=</mo><mtext>trace</mtext><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>What we have really done in this process is one of change of basis. 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xmlns="http://www.w3.org/1998/Math/MathML"><mtext>diag</mtext><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>λ</mi><mi>d</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\text{diag}(\lambda_1, \cdots, \lambda_d)</script><span>. This means that the off-diagonal entries are zero, hence we have found a set of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.176ex" height="1.593ex" role="img" focusable="false" viewBox="0 -694 520 704" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-1640-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></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-1640-TEX-I-1D451"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">d</script><span> orthonormal directions in which the features are decorrelated.</span></p></div></div>
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