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covariance_matrix.html
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</style><title>covariance_matrix</title>
</head>
<body class='typora-export'><div class='typora-export-content'>
<div id='write' class=''><h1 id='covariance-matrix--2d-example'><span>Covariance Matrix | 2D example</span></h1><p><span>Consider a dataset in </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.621ex" height="1.887ex" role="img" focusable="false" viewBox="0 -833.9 1158.6 833.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-1690-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 337V621L237 635L244 648H133Q136 641 137 638T139 603T141 517T141 341Q141 131 140 89T134 37Q133 36 133 35H241ZM457 496Q457 540 449 570T425 615T400 634T377 643Q374 643 339 648Q300 648 281 635Q271 628 270 610T268 481V346H284Q327 346 375 352Q421 364 439 392T457 496ZM492 537T492 496T488 427T478 389T469 371T464 361Q464 360 465 360Q469 360 497 370Q593 400 593 495Q593 592 477 630L457 637L461 626Q474 611 488 561Q492 537 492 496ZM464 243Q411 317 410 317Q404 317 401 315Q384 315 370 312H346L526 35H619L606 50Q553 109 464 243Z"></path><path id="MJX-1690-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="211D" xlink:href="#MJX-1690-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="mn"><use data-c="32" xlink:href="#MJX-1690-TEX-N-32"></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"><mn>2</mn></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbb{R}^{2}</script><span> with the following covariance matrix:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n3" cid="n3" 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="13.719ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 6063.7 2400" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.149ex;"><defs><path id="MJX-1667-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 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46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-1667-TEX-I-1D450" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path><path id="MJX-1667-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-1667-TEX-S3-5D" d="M11 1388V1450H280V-949H11V-887H218V1388H11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr"><g data-mml-node="mtd"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D402" xlink:href="#MJX-1667-TEX-B-1D402"></use></g></g><g data-mml-node="mo" transform="translate(1108.8,0)"><use data-c="3D" xlink:href="#MJX-1667-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(2164.6,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="5B" xlink:href="#MJX-1667-TEX-S3-5B"></use></g><g data-mml-node="mtable" transform="translate(528,0)"><g data-mml-node="mtr" transform="translate(0,700)"><g data-mml-node="mtd"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-1667-TEX-I-1D463"></use></g><g data-mml-node="mn" 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xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">C</mi></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>v</mi><mn>1</mn></msub></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Let </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-1700-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-1700-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-1700-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-1700-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> and </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-1701-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-1701-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1701-TEX-B-1D430"></use></g></g><g data-mml-node="mn" transform="translate(864,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1701-TEX-N-32"></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>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{w}_2</script><span> be the eigenvectors 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-1693-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-1693-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> with eigenvalues </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-1694-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-1694-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-1694-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1694-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> 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-1695-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-1695-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1695-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1695-TEX-N-32"></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>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_2</script><span> respectively, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.63ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 3372.7 844" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1696-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-1696-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-1696-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-1696-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1696-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1696-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1297.3,0)"><use data-c="2265" xlink:href="#MJX-1696-TEX-N-2265"></use></g><g data-mml-node="msub" transform="translate(2353.1,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1696-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1696-TEX-N-32"></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><mo>≥</mo><msub><mi>λ</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\lambda_1 \geq \lambda_2</script><span>. You can think about an image like this:</span></p><p><img src="images/img_covar.png" referrerpolicy="no-referrer"></p><p><span>The variance of the dataset </span><em><span>along</span></em><span> any direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.88ex" height="1.005ex" role="img" focusable="false" viewBox="0 -444 831 444" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-1697-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></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="1D430" xlink:href="#MJX-1697-TEX-B-1D430"></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">w</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{w}</script><span> is given by:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n7" cid="n7" 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="24.541ex" height="6.484ex" role="img" focusable="false" viewBox="0 -1620 10847.2 2865.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-1668-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-1668-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-1668-TEX-N-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-1668-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-1668-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-1668-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-1668-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-1668-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-1668-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-1668-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 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transform="translate(10016.2,0)"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1668-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"><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><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mi>i</mi><mi>T</mi></msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</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">w</mi></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>Using this formula, we can now look at the following variances:</span></p><ul><li><span>Variance along </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.361ex" height="1.344ex" role="img" focusable="false" viewBox="0 -444 1043.6 594" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1698-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 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data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1671-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-1671-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(8931.8,0)"><use data-c="29" xlink:href="#MJX-1671-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(9598.6,0)"><use data-c="3D" xlink:href="#MJX-1671-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(10654.4,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1671-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1671-TEX-N-31"></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><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mn>1</mn><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><mn>1</mn></msub><mo>=</mo><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mn>1</mn><mi>T</mi></msubsup><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>λ</mi><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container></div></div><ul><li><span>Variance along </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-1701-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-1701-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1701-TEX-B-1D430"></use></g></g><g data-mml-node="mn" transform="translate(864,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1701-TEX-N-32"></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>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\mathbf{w}_2</script><span>: This is the second largest eigenvalue.</span></li></ul><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n31" cid="n31" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" 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transform="translate(8931.8,0)"><use data-c="29" xlink:href="#MJX-1672-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(9598.6,0)"><use data-c="3D" xlink:href="#MJX-1672-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(10654.4,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1672-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1672-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><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mn>2</mn><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><mn>2</mn></msub><mo>=</mo><msubsup><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mn>2</mn><mi>T</mi></msubsup><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>2</mn></msub><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">w</mi></mrow><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>λ</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container></div></div><p><span>The trace of the covariance matrix is the sum of the diagonal elements. 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transform="translate(5640.9,0)"><use data-c="2B" xlink:href="#MJX-1673-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(6641.1,0)"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1673-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1673-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"><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><msub><mi>v</mi><mn>2</mn></msub><mo>=</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container></div></div><ul><li><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.936ex" height="1.658ex" role="img" focusable="false" viewBox="0 -583 3065.6 733" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1702-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-1702-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-1702-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 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data-c="2B" xlink:href="#MJX-1702-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(2144,0)"><g data-mml-node="mi"><use data-c="1D463" xlink:href="#MJX-1702-TEX-I-1D463"></use></g><g data-mml-node="mn" transform="translate(518,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1702-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><msub><mi>v</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">v_1 + v_2</script><span> is the total variance of the dataset computed using the standard basis </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.99ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3531.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path 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data-c="1D431" xlink:href="#MJX-1703-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-1703-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1543.6,0)"><use data-c="2C" xlink:href="#MJX-1703-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1988.2,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D431" xlink:href="#MJX-1703-TEX-B-1D431"></use></g></g><g data-mml-node="mn" transform="translate(640,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1703-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(3031.8,0)"><use data-c="7D" xlink:href="#MJX-1703-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">x</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">x</mi></mrow><mn>2</mn></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\{\mathbf{x}_1, \mathbf{x}_2\}</script><span>. </span></li><li><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.379ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 3261.6 844" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-1704-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 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type="math/tex">\lambda_1 + \lambda_2</script><span> is the total variance of the dataset computed using the eigenbasis </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.004ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3979.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-1705-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 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 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-214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="7B" xlink:href="#MJX-1705-TEX-N-7B"></use></g><g data-mml-node="msub" transform="translate(500,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D430" xlink:href="#MJX-1705-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-1705-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1767.6,0)"><use data-c="2C" xlink:href="#MJX-1705-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(2212.2,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use 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As pointed out in an earlier note, the total variance is invariant to the choice of basis. Now, if we represent all the points in an eigenbasis, then the covariance matrix in this setup will be:</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="9.265ex" height="5.43ex" role="img" focusable="false" viewBox="0 -1450 4095.1 2400" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.149ex;"><defs><path id="MJX-1674-TEX-S3-5B" d="M247 -949V1450H516V1388H309V-887H516V-949H247Z"></path><path id="MJX-1674-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-1674-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-1674-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-1674-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-1674-TEX-S3-5D" d="M11 1388V1450H280V-949H11V-887H218V1388H11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="5B" xlink:href="#MJX-1674-TEX-S3-5B"></use></g><g data-mml-node="mtable" transform="translate(528,0)"><g data-mml-node="mtr" transform="translate(0,700)"><g data-mml-node="mtd"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1674-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1674-TEX-N-31"></use></g></g></g><g data-mml-node="mtd" transform="translate(2279.3,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-1674-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-700)"><g data-mml-node="mtd" transform="translate(259.8,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-1674-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(2019.6,0)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D706" xlink:href="#MJX-1674-TEX-I-1D706"></use></g><g data-mml-node="mn" transform="translate(616,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-1674-TEX-N-32"></use></g></g></g></g></g><g data-mml-node="mo" transform="translate(3567.1,0) translate(0 -0.5)"><use data-c="5D" xlink:href="#MJX-1674-TEX-S3-5D"></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="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>λ</mi><mn>1</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>λ</mi><mn>2</mn></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>The covariance matrix in the eigenbasis is diagonal. The diagonal entries represent the variance along the principal components and are the eigenvalues. The off-diagonal entries are zero. Recall that the off diagonal entries measure some kind of correlation between two features. Moving to the eigenbasis de-correlates the features. 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