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class="menu navigation"><div class="list-h"><a target="_blank" rel="external nofollow noopener noreferrer" href="https://github.com/MHuiG" active-action="action-https:githubcomMHuiG"><i class="fa-brands fa-github color-github fa-fw"></i><p>Github</p></a><a href="/pages/friends/" active-action="action-pagesfriends"><i class="fa-duotone fa-link color-friends fa-fw"></i><p>友链</p></a><a href="/pages/about/" active-action="action-pagesabout"><i class="fa-duotone fa-user-tie color-about fa-fw"></i><p>关于</p></a><a href="https://www.travellings.cn/go-by-clouds.html" target="_blank" active-action="action-https:wwwtravellingscngo-by-cloudshtml" rel="external nofollow noopener noreferrer"><i class="fa-duotone fa-subway color-travellings fa-fw"></i><p>Travelling</p></a></div></div></div></div><div id="scroll-down" style="display:none"><i class="fa fa-chevron-down scroll-down-effects"></i></div></div></div><div id="safearea"><div class="body-wrapper"><div id="l_main" class><article itemscope itemtype="http://schema.org/Article" class="article post white-box reveal md shadow floatable blur article-type-post" id="post" itemprop="blogPost"><link itemprop="mainEntityOfPage" href="https://blog.mhuig.top/p/f0765214/"><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="MHuiG"></span><span hidden itemprop="post" itemscope itemtype="http://schema.org/Post"><meta itemprop="name" content="MHuiG"><meta itemprop="description" content="MHuiG&#39;s Blog (MHuiG的博客) MHuiG&#39;s Neverland（MHuiG的梦幻岛） —— MHuiG(@MHuiG) 随便写写画画的地方 - 技术博客"></span><span hidden><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><h1 class="title" itemprop="name headline"> 特征向量和特征值的几何本质</h1><div class="new-meta-box"><div class="new-meta-item author" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a itemprop="url" class="author" target="_blank" href="https://mhuig.top/" rel="nofollow noopener"><img itemprop="image" src="/lib/avatar/avatar-16.webp?v=2199236952" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="><p itemprop="name">MHuiG</p></a></div><div class="new-meta-item wordcount"><a class="notlink"><i class="fa-duotone fa-keyboard fa-fw" aria-hidden="true"></i><p>字数：348 字</p></a></div><div class="new-meta-item readtime"><a class="notlink"><i class="fa-duotone fa-hourglass-half fa-fw" aria-hidden="true"></i><p>时长：1 分钟</p></a></div></div></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p>子矩阵的特征值编码了原矩阵特征向量的隐藏信息。</p><span id="more"></span><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:43.466ex"><svg style="vertical-align:-2.937ex;min-width:43.466ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="7.006ex" role="img" focusable="false"><g stroke="currentColor" 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class="MathJax" jax="SVG"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46 272 46 272 35 272 34 270 22 267 8 264 4T251 0Q249 0 239 0T205 1 141 2Q70 2 50 0H42Q35 7 35 11 37 38 48 46H62Q132 49 164 96 170 102 345 401T523 704Q530 716 547 716H555 572Q578 707 578 706L606 383Q634 60 636 57 641 46 701 46 726 46 726 36 726 34 723 22 720 7 718 4T704 0Q701 0 690 0T651 1 578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260 413 259Q516 259 516 260Z"/></g></g></g></svg></mjx-container>为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D45B" d="M21 287Q22 293 24 303T36 341 56 388 89 425 135 442Q171 442 195 424T225 390 231 369Q231 367 232 367L243 378Q304 442 382 442 436 442 469 415T503 336 465 179 427 52Q427 26 444 26 450 26 453 27 482 32 505 65T540 145Q542 153 560 153 580 153 580 145 580 144 576 130 568 101 554 73T508 17 439-10Q392-10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291 189 157Q156 26 151 16 138-11 108-11 95-11 87-5T76 7 74 17Q74 30 112 180T152 343Q153 348 153 366 153 405 129 405 91 405 66 305 60 285 60 284 58 278 41 278H27Q21 284 21 287Z"/></g></g></g></svg></mjx-container>阶矩阵，若数<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.027ex" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" 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52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g></g></g></svg></mjx-container>，那么数<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.027ex" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644 322 629 373 486T474 207 524 67Q531 47 537 34T546 15 551 6 555 2 556-2 550-11H482Q457 3 450 18T399 152L354 277 340 262Q327 246 293 207T236 141Q211 112 174 69 123 9 111-1T83-12Q47-12 47 20 47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657 166 657 166 673Z"/></g></g></g></svg></mjx-container>称为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46 272 46 272 35 272 34 270 22 267 8 264 4T251 0Q249 0 239 0T205 1 141 2Q70 2 50 0H42Q35 7 35 11 37 38 48 46H62Q132 49 164 96 170 102 345 401T523 704Q530 716 547 716H555 572Q578 707 578 706L606 383Q634 60 636 57 641 46 701 46 726 46 726 36 726 34 723 22 720 7 718 4T704 0Q701 0 690 0T651 1 578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260 413 259Q516 259 516 260Z"/></g></g></g></svg></mjx-container>的特征值，<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442 467 442 494 420T522 361Q522 332 508 314T481 292 458 288Q439 288 427 299T415 328Q415 374 465 391 454 404 425 404 412 404 406 402 368 386 350 336 290 115 290 78 290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145 504 144 502 134 486 77 440 33T333-11Q263-11 227 52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g></g></g></svg></mjx-container>称为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46 272 46 272 35 272 34 270 22 267 8 264 4T251 0Q249 0 239 0T205 1 141 2Q70 2 50 0H42Q35 7 35 11 37 38 48 46H62Q132 49 164 96 170 102 345 401T523 704Q530 716 547 716H555 572Q578 707 578 706L606 383Q634 60 636 57 641 46 701 46 726 46 726 36 726 34 723 22 720 7 718 4T704 0Q701 0 690 0T651 1 578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260 413 259Q516 259 516 260Z"/></g></g></g></svg></mjx-container>的对应于特征值<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.027ex" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644 322 629 373 486T474 207 524 67Q531 47 537 34T546 15 551 6 555 2 556-2 550-11H482Q457 3 450 18T399 152L354 277 340 262Q327 246 293 207T236 141Q211 112 174 69 123 9 111-1T83-12Q47-12 47 20 47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657 166 657 166 673Z"/></g></g></g></svg></mjx-container>的特征向量。</p><p>它的物理意义是：</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:37.281ex"><svg style="vertical-align:-.566ex;min-width:37.281ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="6161 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">一</text></g><g data-mml-node="mi" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">个</text></g><g data-mml-node="mi" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">矩</text></g><g data-mml-node="mi" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">阵</text></g><g data-mml-node="mi" transform="translate(4000,0)"><path data-c="1D434" d="M208 74Q208 50 254 46 272 46 272 35 272 34 270 22 267 8 264 4T251 0Q249 0 239 0T205 1 141 2Q70 2 50 0H42Q35 7 35 11 37 38 48 46H62Q132 49 164 96 170 102 345 401T523 704Q530 716 547 716H555 572Q578 707 578 706L606 383Q634 60 636 57 641 46 701 46 726 46 726 36 726 34 723 22 720 7 718 4T704 0Q701 0 690 0T651 1 578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260 413 259Q516 259 516 260Z"/></g><g data-mml-node="mi" transform="translate(4750,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">乘</text></g><g data-mml-node="mi" transform="translate(5750,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">以</text></g><g data-mml-node="mi" transform="translate(6750,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">一</text></g><g data-mml-node="mi" transform="translate(7750,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">个</text></g><g data-mml-node="mi" transform="translate(8750,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">向</text></g><g data-mml-node="mi" transform="translate(9750,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">量</text></g><g data-mml-node="mi" transform="translate(10750,0)"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442 467 442 494 420T522 361Q522 332 508 314T481 292 458 288Q439 288 427 299T415 328Q415 374 465 391 454 404 425 404 412 404 406 402 368 386 350 336 290 115 290 78 290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145 504 144 502 134 486 77 440 33T333-11Q263-11 227 52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g><g data-mml-node="mi" transform="translate(11322,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:2"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315 301 241Q265 210 201 149L142 93 218 92Q375 92 385 97 392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19 31Q50 38 56 46T86 81Q115 113 136 137 145 147 170 174T204 211 233 244 261 278 284 308 305 340 320 369 333 401 340 431 343 464Q343 527 309 573T212 619Q179 619 154 602T119 569 109 550Q109 549 114 549 132 549 151 535T170 489Q170 464 154 447T109 429Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:41.428ex"><svg style="vertical-align:-.566ex;min-width:41.428ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="7077.5 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">就</text></g><g data-mml-node="mi" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">相</text></g><g data-mml-node="mi" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">当</text></g><g data-mml-node="mi" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">于</text></g><g data-mml-node="mi" transform="translate(4000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">做</text></g><g data-mml-node="mi" transform="translate(5000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">了</text></g><g data-mml-node="mi" transform="translate(6000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">一</text></g><g data-mml-node="mi" transform="translate(7000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">个</text></g><g data-mml-node="mi" transform="translate(8000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">线</text></g><g data-mml-node="mi" transform="translate(9000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">性</text></g><g data-mml-node="mi" transform="translate(10000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">变</text></g><g data-mml-node="mi" transform="translate(11000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">换</text></g><g data-mml-node="mi" transform="translate(12000,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644 322 629 373 486T474 207 524 67Q531 47 537 34T546 15 551 6 555 2 556-2 550-11H482Q457 3 450 18T399 152L354 277 340 262Q327 246 293 207T236 141Q211 112 174 69 123 9 111-1T83-12Q47-12 47 20 47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657 166 657 166 673Z"/></g><g data-mml-node="mi" transform="translate(12583,0)"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442 467 442 494 420T522 361Q522 332 508 314T481 292 458 288Q439 288 427 299T415 328Q415 374 465 391 454 404 425 404 412 404 406 402 368 386 350 336 290 115 290 78 290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145 504 144 502 134 486 77 440 33T333-11Q263-11 227 52 186-10 133-10H127Q78-10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101 142 81 130 66T107 46 94 41L91 40Q91 39 97 36T113 29 132 26Q168 26 194 71 203 87 217 139T245 247 261 313Q266 340 266 352 266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"/></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(13155,0)"><g data-mml-node="mo"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</text></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:3"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 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186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:29.765ex"><svg style="vertical-align:-.566ex;min-width:29.765ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="4500 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">方</text></g><g data-mml-node="mi" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">向</text></g><g data-mml-node="mi" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">仍</text></g><g data-mml-node="mi" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">然</text></g><g data-mml-node="mi" transform="translate(4000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">保</text></g><g data-mml-node="mi" transform="translate(5000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">持</text></g><g data-mml-node="mi" transform="translate(6000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">不</text></g><g data-mml-node="mi" transform="translate(7000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">变</text></g><g data-mml-node="mi" transform="translate(8000,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:4"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="34" d="M462 0Q444 3 333 3 217 3 199 0H190V46H221Q241 46 248 46T265 48 279 53 286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675 336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212 183 211H293Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:40.133ex"><svg style="vertical-align:-.566ex;min-width:40.133ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="6791.5 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">只</text></g><g data-mml-node="mi" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">是</text></g><g data-mml-node="mi" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">拉</text></g><g data-mml-node="mi" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">伸</text></g><g data-mml-node="mi" transform="translate(4000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">或</text></g><g data-mml-node="mi" transform="translate(5000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">者</text></g><g data-mml-node="mi" transform="translate(6000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">压</text></g><g data-mml-node="mi" transform="translate(7000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">缩</text></g><g data-mml-node="mi" transform="translate(8000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">一</text></g><g data-mml-node="mi" transform="translate(9000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">定</text></g><g data-mml-node="mi" transform="translate(10000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">倍</text></g><g data-mml-node="mi" transform="translate(11000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g><g data-mml-node="mi" transform="translate(12000,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644 322 629 373 486T474 207 524 67Q531 47 537 34T546 15 551 6 555 2 556-2 550-11H482Q457 3 450 18T399 152L354 277 340 262Q327 246 293 207T236 141Q211 112 174 69 123 9 111-1T83-12Q47-12 47 20 47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657 166 657 166 673Z"/></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(12583,0)"><g data-mml-node="mo"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</text></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:5"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22 294 22 326 82 345 115 345 210 345 313 318 349 292 382 260 382H254Q176 382 136 314 132 307 129 306T114 304Q97 304 95 310 93 314 93 485V614Q93 664 98 664 100 666 102 666 103 666 123 658T178 642 253 634Q324 634 389 662 397 666 402 666 410 666 410 648V635Q328 538 205 538 174 538 149 544L139 546V374Q158 388 169 396T205 412 256 420Q337 420 393 355T449 201Q449 109 385 44T229-22Q148-22 99 32T50 154Q50 178 61 192T84 210 107 214Q132 214 148 197T164 157Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>特征向量和特征值的几何本质，其实就是：</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:34.29ex"><svg style="vertical-align:-.566ex;min-width:34.29ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveAspectRatio="xMidYMid" viewBox="5500 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">空</text></g><g data-mml-node="mi" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">间</text></g><g data-mml-node="mi" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">矢</text></g><g data-mml-node="mi" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">量</text></g><g data-mml-node="mi" transform="translate(4000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">的</text></g><g data-mml-node="mi" transform="translate(5000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">旋</text></g><g data-mml-node="mi" transform="translate(6000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">转</text></g><g data-mml-node="mi" transform="translate(7000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">和</text></g><g data-mml-node="mi" transform="translate(8000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">缩</text></g><g data-mml-node="mi" transform="translate(9000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">放</text></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(10000,0)"><g data-mml-node="mo"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</text></g></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:6"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601 351 623 311 626 240 626 194 566 147 500 147 364L148 360Q153 366 156 373 197 433 263 433H267Q313 433 348 414 372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149 440 90 387 34T253-22Q225-22 199-14T143 16 92 75 56 172 42 313ZM257 397Q227 397 205 380T171 335 154 278 148 216Q148 133 160 97T198 39Q222 21 251 21 302 21 329 59 342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><img src="https://static.mhuig.top/npm/mhgoos@0.0.1652000248732/20191117100041.webp" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1652000248732/20191117100041.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="线性变换"></p><p>线性变换 A 对于特征空间只起到 “扩张 (或者压缩)” 的作用（扩张后还是同样的特征空间）</p><div class="story post-story"><h2 id="求解特征向量"><a href="#求解特征向量" class="headerlink" title="求解特征向量"></a>求解特征向量</h2><p>按照传统解法：</p><p>计算特征多项式→求解特征值→求解齐次线性方程组，得出特征向量。</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:50.125ex"><svg style="vertical-align:-.566ex;min-width:50.125ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math"><g data-mml-node="mtable" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 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302L274-226Q180-141 137-14T94 250Z"/><path data-c="37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646 364 644H485V605L417 512Q408 500 387 472T360 435 339 403 319 367 305 330 292 284 284 230 278 162 275 80Q275 66 275 52T274 28V19Q270 2 255-10T221-22Q210-22 200-19T179 0 168 40Q168 198 265 368 285 400 349 489L395 552H302Q128 552 119 546 113 543 108 522T98 479L95 458V455H55V458Z" transform="translate(389,0)"/><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z" transform="translate(889,0)"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>全新的方法：</p><p><mjx-container class="MathJax" jax="SVG" display="true" width="full" style="min-width:43.466ex"><svg style="vertical-align:-2.937ex;min-width:43.466ex" xmlns="http://www.w3.org/2000/svg" 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361Q249 344 246 335 246 329 231 291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="mi" transform="translate(518,-150) scale(0.707)"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369 98 420 158 442Q197 442 223 419T250 357Q250 340 236 301T196 196 154 83Q149 61 149 51 149 26 166 26 175 26 185 29T208 43 235 78 260 137Q263 149 265 151T282 153Q302 153 302 143 302 135 293 112T268 61 223 11 161-11Q129-11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287Z"/></g></g><g data-mml-node="msub" transform="translate(1090,0)"><g 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data-id-align="true"/><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/><path data-c="38" d="M70 417T70 494 124 618 248 666Q319 666 374 624T429 515Q429 485 418 459T392 417 361 389 335 371 324 363L338 354Q352 344 366 334T382 323Q457 264 457 174 457 95 399 37T249-22Q159-22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368 70 417 70 494ZM286 386 292 390Q298 394 301 396T311 403 323 413 334 425 345 438 355 454 364 471 369 491 371 513Q371 556 342 586T275 624Q268 625 242 625 201 625 165 599T128 534Q128 511 141 492T167 463 217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194 353 216 330 234 301 253 274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239 107 157Q107 98 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291T200 202 182 113Q182 86 187 69 200 26 250 26 287 26 319 60T369 139 398 222 409 277Q409 300 401 317T383 343 365 361 357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159 347 40 241-11Q177-11 139 22 102 54 102 117 102 148 110 181T151 298Q173 362 173 380Z"/></g><g data-mml-node="mi" transform="translate(518,-150) scale(0.707)"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369 98 420 158 442Q197 442 223 419T250 357Q250 340 236 301T196 196 154 83Q149 61 149 51 149 26 166 26 175 26 185 29T208 43 235 78 260 137Q263 149 265 151T282 153Q302 153 302 143 302 135 293 112T268 61 223 11 161-11Q129-11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287Z"/></g></g><g data-mml-node="msub" transform="translate(812,0)"><g data-mml-node="mo"><path data-c="2C" d="M78 35T78 60 94 103 137 121Q165 121 187 96T210 8Q210-27 201-60T180-117 154-158 130-185 117-194Q113-194 104-185T95-172Q95-168 106-156T131-126 157-76 173-3V9L172 8Q170 7 167 6T161 3 152 1 140 0Q113 0 96 17Z"/></g><g data-mml-node="mi" transform="translate(311,-150) scale(0.707)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405 240 405 220 393T185 362 161 325 144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432 341 418 354 401T367 348V332L318 133Q267-67 264-75 246-125 194-164T75-204Q25-204 7-183T-12-137Q-12-110 7-91T53-71Q70-71 82-81T95-112Q95-148 63-167 69-168 77-168 111-168 139-140T182-74L193-32Q204 11 219 72T251 197 278 308 289 365Q289 372 288 376Z"/></g></g></g></g></svg></mjx-container>为特征值<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.357ex" xmlns="http://www.w3.org/2000/svg" width="2.059ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 910 851.8"><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"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644 322 629 373 486T474 207 524 67Q531 47 537 34T546 15 551 6 555 2 556-2 550-11H482Q457 3 450 18T399 152L354 277 340 262Q327 246 293 207T236 141Q211 112 174 69 123 9 111-1T83-12Q47-12 47 20 47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657 166 657 166 673Z"/></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369 98 420 158 442Q197 442 223 419T250 357Q250 340 236 301T196 196 154 83Q149 61 149 51 149 26 166 26 175 26 185 29T208 43 235 78 260 137Q263 149 265 151T282 153Q302 153 302 143 302 135 293 112T268 61 223 11 161-11Q129-11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287Z"/></g></g></g></g></svg></mjx-container>对应特征向量的第<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.462ex" xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405 240 405 220 393T185 362 161 325 144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432 341 418 354 401T367 348V332L318 133Q267-67 264-75 246-125 194-164T75-204Q25-204 7-183T-12-137Q-12-110 7-91T53-71Q70-71 82-81T95-112Q95-148 63-167 69-168 77-168 111-168 139-140T182-74L193-32Q204 11 219 72T251 197 278 308 289 365Q289 372 288 376Z"/></g></g></g></svg></mjx-container>个元素；</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="5.516ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2438 1000"><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"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644 322 629 373 486T474 207 524 67Q531 47 537 34T546 15 551 6 555 2 556-2 550-11H482Q457 3 450 18T399 152L354 277 340 262Q327 246 293 207T236 141Q211 112 174 69 123 9 111-1T83-12Q47-12 47 20 47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657 166 657 166 673Z"/></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369 98 420 158 442Q197 442 223 419T250 357Q250 340 236 301T196 196 154 83Q149 61 149 51 149 26 166 26 175 26 185 29T208 43 235 78 260 137Q263 149 265 151T282 153Q302 153 302 143 302 135 293 112T268 61 223 11 161-11Q129-11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287Z"/></g></g><g data-mml-node="mo" transform="translate(910,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/></g><g data-mml-node="mi" transform="translate(1299,0)"><path data-c="1D434" d="M208 74Q208 50 254 46 272 46 272 35 272 34 270 22 267 8 264 4T251 0Q249 0 239 0T205 1 141 2Q70 2 50 0H42Q35 7 35 11 37 38 48 46H62Q132 49 164 96 170 102 345 401T523 704Q530 716 547 716H555 572Q578 707 578 706L606 383Q634 60 636 57 641 46 701 46 726 46 726 36 726 34 723 22 720 7 718 4T704 0Q701 0 690 0T651 1 578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260 413 259Q516 259 516 260Z"/></g><g data-mml-node="mo" transform="translate(2049,0)"><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z"/></g></g></g></svg></mjx-container> 为矩阵<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46 272 46 272 35 272 34 270 22 267 8 264 4T251 0Q249 0 239 0T205 1 141 2Q70 2 50 0H42Q35 7 35 11 37 38 48 46H62Q132 49 164 96 170 102 345 401T523 704Q530 716 547 716H555 572Q578 707 578 706L606 383Q634 60 636 57 641 46 701 46 726 46 726 36 726 34 723 22 720 7 718 4T704 0Q701 0 690 0T651 1 578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260 413 259Q516 259 516 260Z"/></g></g></g></svg></mjx-container>的第<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369 98 420 158 442Q197 442 223 419T250 357Q250 340 236 301T196 196 154 83Q149 61 149 51 149 26 166 26 175 26 185 29T208 43 235 78 260 137Q263 149 265 151T282 153Q302 153 302 143 302 135 293 112T268 61 223 11 161-11Q129-11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281 56 279 53 278 49 278 41 278H27Q21 284 21 287Z"/></g></g></g></svg></mjx-container>个特征向量；</p><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.666ex" xmlns="http://www.w3.org/2000/svg" width="3.041ex" height="2.211ex" role="img" focusable="false" viewBox="0 -683 1344.3 977.2"><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"><path data-c="1D440" d="M289 629Q289 635 232 637 208 637 201 638T194 648Q194 649 196 659 197 662 198 666T199 671 201 676 203 679 207 681 212 683 220 683 232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117 510 154Q534 190 574 254T662 394Q837 673 839 675 840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673 1051 668 1048 656T1045 643Q1041 637 1008 637 968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50 822 48 855 46H886Q892 37 892 35 892 19 885 5 880 0 869 0 864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11 592 13 594 25 598 41 602 43T625 46Q652 46 685 49 699 52 704 61 706 65 742 207T813 490 848 631L654 322Q458 10 453 5 451 4 449 3 444 0 433 0 418 0 415 7 413 11 374 317L335 624 267 354Q200 88 200 79 206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1 274 0 267 0 265 0 255 0T221 1 157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46 135 46 154 86 158 92 223 354T289 629Z"/></g><g data-mml-node="mi" transform="translate(1003,-150) scale(0.707)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405 240 405 220 393T185 362 161 325 144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432 341 418 354 401T367 348V332L318 133Q267-67 264-75 246-125 194-164T75-204Q25-204 7-183T-12-137Q-12-110 7-91T53-71Q70-71 82-81T95-112Q95-148 63-167 69-168 77-168 111-168 139-140T182-74L193-32Q204 11 219 72T251 197 278 308 289 365Q289 372 288 376Z"/></g></g></g></g></svg></mjx-container> 为矩阵<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46 272 46 272 35 272 34 270 22 267 8 264 4T251 0Q249 0 239 0T205 1 141 2Q70 2 50 0H42Q35 7 35 11 37 38 48 46H62Q132 49 164 96 170 102 345 401T523 704Q530 716 547 716H555 572Q578 707 578 706L606 383Q634 60 636 57 641 46 701 46 726 46 726 36 726 34 723 22 720 7 718 4T704 0Q701 0 690 0T651 1 578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260 413 259Q516 259 516 260Z"/></g></g></g></svg></mjx-container>的第<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.462ex" xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.957ex" role="img" focusable="false" viewBox="0 -661 412 865"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405 240 405 220 393T185 362 161 325 144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432 341 418 354 401T367 348V332L318 133Q267-67 264-75 246-125 194-164T75-204Q25-204 7-183T-12-137Q-12-110 7-91T53-71Q70-71 82-81T95-112Q95-148 63-167 69-168 77-168 111-168 139-140T182-74L193-32Q204 11 219 72T251 197 278 308 289 365Q289 372 288 376Z"/></g></g></g></svg></mjx-container>个余子式，<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.666ex" xmlns="http://www.w3.org/2000/svg" width="7.142ex" height="2.363ex" role="img" focusable="false" viewBox="0 -750 3156.7 1044.2"><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"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644 322 629 373 486T474 207 524 67Q531 47 537 34T546 15 551 6 555 2 556-2 550-11H482Q457 3 450 18T399 152L354 277 340 262Q327 246 293 207T236 141Q211 112 174 69 123 9 111-1T83-12Q47-12 47 20 47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657 166 657 166 673Z"/></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686 294 679 244 477 194 279 194 272 213 282 223 291 247 309 292 354T362 415Q402 442 438 442 468 442 485 423T503 369Q503 344 496 327T477 302 456 291 438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316 228 255Q230 254 243 252T267 246 293 238 320 224 342 206 359 180 365 147Q365 130 360 106T354 66Q354 26 381 26 429 26 459 145 461 153 479 153H483Q499 153 499 144 499 139 496 130 455-11 378-11 333-11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206 200 217 182 220H180Q168 178 159 139T145 81 136 44 129 20 122 7 111-2Q98-11 83-11 66-11 57-1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640 121 647Z"/></g></g><g data-mml-node="mo" transform="translate(1034.4,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488 164 576 202 643 244 695 277 729 302 750H315 319Q333 750 333 741 333 738 316 720T275 667 226 581 184 443 167 250 184 58 225-81 274-167 316-220 333-241Q333-250 318-250H315 302L274-226Q180-141 137-14T94 250Z"/></g><g data-mml-node="msub" transform="translate(1423.4,0)"><g data-mml-node="mi"><path data-c="1D440" d="M289 629Q289 635 232 637 208 637 201 638T194 648Q194 649 196 659 197 662 198 666T199 671 201 676 203 679 207 681 212 683 220 683 232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117 510 154Q534 190 574 254T662 394Q837 673 839 675 840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673 1051 668 1048 656T1045 643Q1041 637 1008 637 968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50 822 48 855 46H886Q892 37 892 35 892 19 885 5 880 0 869 0 864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11 592 13 594 25 598 41 602 43T625 46Q652 46 685 49 699 52 704 61 706 65 742 207T813 490 848 631L654 322Q458 10 453 5 451 4 449 3 444 0 433 0 418 0 415 7 413 11 374 317L335 624 267 354Q200 88 200 79 206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1 274 0 267 0 265 0 255 0T221 1 157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46 135 46 154 86 158 92 223 354T289 629Z"/></g><g data-mml-node="mi" transform="translate(1003,-150) scale(0.707)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405 240 405 220 393T185 362 161 325 144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432 341 418 354 401T367 348V332L318 133Q267-67 264-75 246-125 194-164T75-204Q25-204 7-183T-12-137Q-12-110 7-91T53-71Q70-71 82-81T95-112Q95-148 63-167 69-168 77-168 111-168 139-140T182-74L193-32Q204 11 219 72T251 197 278 308 289 365Q289 372 288 376Z"/></g></g><g data-mml-node="mo" transform="translate(2767.7,0)"><path data-c="29" d="M60 749 64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12 224-76 186-143 145-194 113-227 90-246Q87-249 86-250H74Q66-250 63-250T58-247 55-238Q56-237 66-225 221-64 221 250T66 725Q56 737 55 738 55 746 60 749Z"/></g></g></g></svg></mjx-container>是该主子式的第<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686 294 679 244 477 194 279 194 272 213 282 223 291 247 309 292 354T362 415Q402 442 438 442 468 442 485 423T503 369Q503 344 496 327T477 302 456 291 438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316 228 255Q230 254 243 252T267 246 293 238 320 224 342 206 359 180 365 147Q365 130 360 106T354 66Q354 26 381 26 429 26 459 145 461 153 479 153H483Q499 153 499 144 499 139 496 130 455-11 378-11 333-11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206 200 217 182 220H180Q168 178 159 139T145 81 136 44 129 20 122 7 111-2Q98-11 83-11 66-11 57-1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640 121 647Z"/></g></g></g></svg></mjx-container>个特征值.</p><p>通过删除原始矩阵的行和列，创建子矩阵。</p><p>子矩阵和原始矩阵的特征值组合在一起，就可以计算原始矩阵的特征向量。</p><p>简而言之，已知特征值，一个方程式就可以求得特征向量。</p></div><div class="story post-story"><h2 id="参考文献"><a href="#参考文献" class="headerlink" title="参考文献"></a>参考文献</h2><p><a target="_blank" rel="external nofollow noopener noreferrer" href="https://arxiv.org/abs/1908.03795">Eigenvectors from Eigenvalues</a></p><p><a target="_blank" rel="external nofollow noopener noreferrer" href="https://arxiv.org/abs/1907.02534">Eigenvalues: the Rosetta Stone for Neutrino Oscillations in Matter</a></p></div></div><div class="footer"><div class="copyright license"><div class="license-title">特征向量和特征值的几何本质</div><div class="license-link"><a href="https://blog.mhuig.top/p/f0765214/">https://blog.mhuig.top/p/f0765214/</a></div><div class="license-meta"><div class="license-meta-item"><div class="license-meta-title">本文作者</div><div class="license-meta-text">MHuiG</div></div><div class="license-meta-item"><div class="license-meta-title">发布于</div><div class="license-meta-text">2019年11月17日</div></div><div class="license-meta-item"><div class="license-meta-title">许可协议</div><div class="license-meta-text"><a href="https://creativecommons.org/licenses/by-nc-sa/4.0/deed.zh#" target="_blank" rel="external nofollow noopener noreferrer">CC BY-NC-SA 4.0</a></div></div></div><div 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