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href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>特征向量和特征值的几何本质 - Magicland</title><meta name="keywords" content="数学,线性代数, Math,特征向量,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc name="description" content="本文探讨特征值(λ)与特征向量(x)的定义(Ax=λx)，阐释其几何本质为空间向量线性变换中保持方向的缩放（旋转与伸缩），物理意义为矩阵作用下向量仅拉伸/压缩λ倍，介绍传统求解步骤（计算特征多项式、求解特征值、解方程组）及基于子矩阵特征值编码原矩阵隐藏信息的新公式|v_i,j|²计算特征向量元素的方法。 - MHuiG - Magicland"><meta property="og:type" content="article"><meta property="og:title" content="特征向量和特征值的几何本质"><meta property="og:url" content="https://blog.mhuig.top/p/f0765214/"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="本文探讨特征值(λ)与特征向量(x)的定义(Ax=λx)，阐释其几何本质为空间向量线性变换中保持方向的缩放（旋转与伸缩），物理意义为矩阵作用下向量仅拉伸/压缩λ倍，介绍传统求解步骤（计算特征多项式、求解特征值、解方程组）及基于子矩阵特征值编码原矩阵隐藏信息的新公式|v_i,j|²计算特征向量元素的方法。"><meta property="og:locale"><meta property="og:image" 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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="Magicland"></span><span hidden itemprop="post" itemscope itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="特征向量和特征值的几何本质"><meta itemprop="description" content="本文探讨特征值(λ)与特征向量(x)的定义(Ax=λx)，阐释其几何本质为空间向量线性变换中保持方向的缩放（旋转与伸缩），物理意义为矩阵作用下向量仅拉伸/压缩λ倍，介绍传统求解步骤（计算特征多项式、求解特征值、解方程组）及基于子矩阵特征值编码原矩阵隐藏信息的新公式|v_i,j|²计算特征向量元素的方法。"></span><div class="article-meta" id="top"><h1 class="title allow-select" 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" alt="author"><img itemprop="image" src="/lib/avatar/avatar-16.webp" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="author"><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><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 43.466ex;"><svg style="vertical-align: -2.895ex; min-width: 43.466ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="6.921ex" role="img" focusable="false" data-mjx-viewbox="0 -1779.5 19212 3058.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1779.5)"><g data-mml-node="math" data-latex=" |v_i,_j|^2=\frac{\prod_{k=1}^{n-1}(λ_i(A)-λ_k(M_j))}{\prod_{k=1,k\not=i}^{n-1}(λ_i(A)-λ_k(A))}"><g data-mml-node="mtable" data-latex="|v_i,_j|^2 =\frac{\prod_{k=1}^{n-1}(λ_i(A)-λ_k(M_j))}{\prod_{k=1,k\not=i}^{n-1}(λ_i(A)-λ_k(A))}" transform="translate(2078,0) 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19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g></g></g></svg></mjx-container>阶矩阵，若数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.029ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.6ex" role="img" focusable="false" viewbox="0 -694 583 707"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="λ"><g data-mml-node="mi" data-latex="λ"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g></g></g></svg></mjx-container>和<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" 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" data-latex="n"><g data-mml-node="mi" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18C427 18 420 28 420 47C420 61 426 84 438 115C478 224 498 296 498 333C498 403 451 442 381 442C322 442 271 416 230 363C222 407 187 442 136 442C80 442 58 390 44 345C34 313 29 294 29 287C29 278 34 273 45 273C50 273 53 274 56 276C61 285 64 292 65 299C83 375 106 413 133 413C151 413 160 399 160 371C160 358 155 331 144 290L87 63C84 50 78 24 78 19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g></g></g></svg></mjx-container>维非<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 500 688"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="0"><g data-mml-node="mn" data-latex="0"><path data-c="30" d="M249-22C390-22 460 92 460 320C460 473 428 575 365 625C330 652 291 666 250 666C109 666 39 551 39 320C39 136 88-22 249-22M361 524C368 489 371 425 371 332C371 240 367 172 360 128C347 48 310 8 249 8C226 8 203 17 182 34C155 57 139 104 132 176C129 201 128 253 128 332C128 419 131 480 136 513C145 568 163 603 191 618C213 630 232 636 249 636C314 636 350 583 361 524Z"/></g></g></g></svg></mjx-container>列向量<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" 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" data-latex="x"><g data-mml-node="mi" data-latex="x"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g></g></g></svg></mjx-container>满足<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.991ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1322 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="Ax=λx"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g><g data-mml-node="mi" data-latex="x" transform="translate(750,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.002ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 2210.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="Ax=λx"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mi" data-latex="λ" transform="translate(1055.8,0)"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g><g data-mml-node="mi" data-latex="x" transform="translate(1638.8,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g></g></g></svg></mjx-container>，那么数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.029ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.6ex" role="img" focusable="false" viewbox="0 -694 583 707"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="λ"><g data-mml-node="mi" data-latex="λ"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g></g></g></svg></mjx-container>称为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><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" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g></g></g></svg></mjx-container>的特征值，<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" 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" data-latex="x"><g data-mml-node="mi" data-latex="x"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g></g></g></svg></mjx-container>称为<mjx-container class="MathJax" jax="SVG" overflow="overflow"><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" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g></g></g></svg></mjx-container>的对应于特征值<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.029ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.6ex" role="img" focusable="false" viewbox="0 -694 583 707"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="λ"><g data-mml-node="mi" data-latex="λ"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></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" overflow="overflow" display="true" width="full" style="min-width: 37.281ex;"><svg style="vertical-align: -0.566ex; min-width: 37.281ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 16478 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="一个矩阵A乘以一个向量x，"><g data-mml-node="mtable" data-latex="一个矩阵A乘以一个向量x，" 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" data-latex="一"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">一</text></g><g data-mml-node="mi" data-latex="个" 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" data-latex="矩" 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" data-latex="阵" 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" data-latex="A" transform="translate(4000,0)"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g><g data-mml-node="mi" data-latex="乘" 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" data-latex="以" 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" data-latex="一" 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" data-latex="个" 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" data-latex="向" 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" data-latex="量" 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" data-latex="x" transform="translate(10750,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g><g data-mml-node="mi" data-latex="，" 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" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{2}" transform="translate(389,0)"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 41.428ex;"><svg style="vertical-align: -0.566ex; min-width: 41.428ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 18311 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="就相当于做了一个线性变换λx。"><g data-mml-node="mtable" data-latex="就相当于做了一个线性变换λx。" 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" data-latex="就"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">就</text></g><g data-mml-node="mi" data-latex="相" 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" data-latex="当" 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" data-latex="于" 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" data-latex="做" 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" data-latex="了" 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" data-latex="一" 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" data-latex="个" 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" data-latex="线" 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" data-latex="性" 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" data-latex="变" 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" data-latex="换" 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" data-latex="λ" transform="translate(12000,0)"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g><g data-mml-node="mi" data-latex="x" transform="translate(12583,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g><g data-mml-node="mo" data-latex="。" transform="translate(13155,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</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:3"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{3}" transform="translate(389,0)"><path data-c="33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 29.765ex;"><svg style="vertical-align: -0.566ex; min-width: 29.765ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 13156 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="方向仍然保持不变，"><g data-mml-node="mtable" data-latex="方向仍然保持不变，" 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" data-latex="方"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">方</text></g><g data-mml-node="mi" data-latex="向" 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" data-latex="仍" 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" data-latex="然" 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" data-latex="保" 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" data-latex="持" 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" data-latex="不" 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" data-latex="变" 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" data-latex="，" 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" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{4}" transform="translate(389,0)"><path data-c="34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 40.133ex;"><svg style="vertical-align: -0.566ex; min-width: 40.133ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 17739 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="只是拉伸或者压缩一定倍数λ。"><g data-mml-node="mtable" data-latex="只是拉伸或者压缩一定倍数λ。" 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" data-latex="只"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">只</text></g><g data-mml-node="mi" data-latex="是" 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" data-latex="拉" 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" data-latex="伸" 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" data-latex="或" 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" data-latex="者" 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" data-latex="压" 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" data-latex="缩" 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" data-latex="一" 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" data-latex="定" 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" data-latex="倍" 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" data-latex="数" 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" data-latex="λ" transform="translate(12000,0)"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g><g data-mml-node="mo" data-latex="。" transform="translate(12583,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</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:5"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{5}" transform="translate(389,0)"><path data-c="35" d="M118 315C123 315 129 319 134 326C164 371 205 393 257 393C292 393 319 373 337 332C348 305 354 264 354 209C354 146 346 102 331 76C306 35 272 14 229 14C162 14 109 62 91 114C94 113 96 114 100 114C130 114 155 137 155 167C155 198 130 219 100 219C65 219 50 200 50 163C50 63 131-22 231-22C292-22 344 0 386 44C428 88 449 141 449 202C449 260 432 310 398 353C361 400 315 423 259 423C212 423 171 408 138 378L138 556C165 548 191 544 218 544C264 544 304 555 338 578C369 597 390 616 402 634C408 642 411 648 411 651C411 661 406 666 396 666C341 645 294 634 256 634C211 634 168 643 127 662C122 664 118 665 114 665C105 665 100 656 100 637L100 345C100 324 100 315 118 315Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></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" overflow="overflow" display="true" width="full" style="min-width: 34.29ex;"><svg style="vertical-align: -0.566ex; min-width: 34.29ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 15156 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="空间矢量的旋转和缩放。"><g data-mml-node="mtable" data-latex="空间矢量的旋转和缩放。" 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" data-latex="空"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">空</text></g><g data-mml-node="mi" data-latex="间" 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" data-latex="矢" 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" data-latex="量" 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" data-latex="的" 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" data-latex="旋" 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" data-latex="转" 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" data-latex="和" 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" data-latex="缩" 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" data-latex="放" 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="mo" data-latex="。" transform="translate(10000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">。</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:6"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{6}" transform="translate(389,0)"><path data-c="36" d="M383 504C416 504 432 521 432 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43.466ex;"><svg style="vertical-align: -2.895ex; min-width: 43.466ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="6.921ex" role="img" focusable="false" data-mjx-viewbox="0 -1779.5 19212 3058.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1779.5)"><g data-mml-node="math" data-latex=" |v_i,_j|^2=\frac{\prod_{k=1}^{n-1}(λ_i(A)-λ_k(M_j))}{\prod_{k=1,k\not=i}^{n-1}(λ_i(A)-λ_k(A))}"><g data-mml-node="mtable" data-latex="|v_i,_j|^2 =\frac{\prod_{k=1}^{n-1}(λ_i(A)-λ_k(M_j))}{\prod_{k=1,k\not=i}^{n-1}(λ_i(A)-λ_k(A))}" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1779.5) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveaspectratio="xMidYMid" viewbox="7528 -1779.5 1 3058.9"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,76.5)"><g data-mml-node="mtd"><g data-mml-node="mo" data-latex="|"><path data-c="7C" d="M139-250C152-250 159-242 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0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(9880.9,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(10269.9,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 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transform="translate(389,0)"><path data-c="38" d="M250 666C201 666 158 650 122 618C86 586 68 546 68 497C68 456 83 419 113 386C120 378 142 361 179 335C88 288 42 227 42 153C42 100 64 57 107 24C147-7 194-22 249-22C305-22 354-4 395 32C436 68 457 115 457 170C457 216 441 257 408 294C396 307 365 330 316 361C393 402 431 454 431 515C431 606 344 666 250 666M379 515C379 463 348 418 286 381L167 459C136 479 120 505 120 536C120 596 185 633 249 633C318 633 379 584 379 515M250 14C168 14 99 73 99 153C99 220 136 274 210 315L328 240C376 209 400 174 400 134C400 62 325 14 250 14Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 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135-181 152-158C186-110 203-57 203 0C203 53 185 106 139 106Z"/></g><g data-mml-node="mi" transform="translate(311,-150) scale(0.707)" data-latex="j"><path data-c="1D457" d="M397 621C397 648 383 661 356 661C328 661 300 633 300 605C300 578 313 565 340 565C368 565 397 593 397 621M267 444C214 444 169 404 147 369C120 326 107 299 107 287C107 278 112 274 122 274C127 274 130 275 133 276C138 284 141 291 144 296C176 375 216 414 264 414C282 414 291 400 291 373C291 358 289 341 284 323L192-46C178-104 138-175 75-175C66-175 57-174 48-171C73-161 85-143 85-118C85-92 71-79 44-79C12-79-13-108-13-140C-13-184 30-205 77-205C120-205 160-190 197-160C232-131 254-94 265-51L356 311C359 324 361 337 361 349C361 404 322 444 267 444Z"/></g></g></g></g></svg></mjx-container>为特征值<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.357ex;" 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" data-latex="λ_i"><g data-mml-node="msub" data-latex="λ_i"><g data-mml-node="mi" data-latex="λ"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)" data-latex="i"><path data-c="1D456" d="M284 621C284 648 271 661 244 661C216 661 188 633 188 605C188 578 202 565 229 565C257 565 284 593 284 621M259 138C237 59 205 19 164 19C151 19 144 28 144 47C144 64 173 150 232 306C240 329 244 347 244 360C244 409 210 445 161 445C118 445 84 420 59 369C39 328 29 301 29 288C29 279 34 275 45 275C58 275 60 281 64 295C87 375 118 415 158 415C171 415 178 406 178 387C178 373 175 357 168 338C145 275 121 208 101 155C86 115 78 88 78 74C78 25 114-11 162-11C205-11 239 14 264 64C283 103 293 130 293 145C293 154 288 159 277 159C274 159 259 150 259 138Z"/></g></g></g></g></svg></mjx-container>对应特征向量的第<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.959ex" role="img" focusable="false" viewbox="0 -661 412 866"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="j"><g data-mml-node="mi" data-latex="j"><path data-c="1D457" d="M397 621C397 648 383 661 356 661C328 661 300 633 300 605C300 578 313 565 340 565C368 565 397 593 397 621M267 444C214 444 169 404 147 369C120 326 107 299 107 287C107 278 112 274 122 274C127 274 130 275 133 276C138 284 141 291 144 296C176 375 216 414 264 414C282 414 291 400 291 373C291 358 289 341 284 323L192-46C178-104 138-175 75-175C66-175 57-174 48-171C73-161 85-143 85-118C85-92 71-79 44-79C12-79-13-108-13-140C-13-184 30-205 77-205C120-205 160-190 197-160C232-131 254-94 265-51L356 311C359 324 361 337 361 349C361 404 322 444 267 444Z"/></g></g></g></svg></mjx-container>个元素;</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.561ex;" xmlns="http://www.w3.org/2000/svg" width="5.516ex" height="2.253ex" role="img" focusable="false" viewbox="0 -748 2438 996"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="λ_i(A)"><g data-mml-node="msub" data-latex="λ_i"><g data-mml-node="mi" data-latex="λ"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 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748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></mjx-container> 为矩阵<mjx-container class="MathJax" jax="SVG" overflow="overflow"><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" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g></g></g></svg></mjx-container>的第<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" 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" data-latex="i"><g data-mml-node="mi" data-latex="i"><path data-c="1D456" d="M284 621C284 648 271 661 244 661C216 661 188 633 188 605C188 578 202 565 229 565C257 565 284 593 284 621M259 138C237 59 205 19 164 19C151 19 144 28 144 47C144 64 173 150 232 306C240 329 244 347 244 360C244 409 210 445 161 445C118 445 84 420 59 369C39 328 29 301 29 288C29 279 34 275 45 275C58 275 60 281 64 295C87 375 118 415 158 415C171 415 178 406 178 387C178 373 175 357 168 338C145 275 121 208 101 155C86 115 78 88 78 74C78 25 114-11 162-11C205-11 239 14 264 64C283 103 293 130 293 145C293 154 288 159 277 159C274 159 259 150 259 138Z"/></g></g></g></svg></mjx-container>个特征向量;</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.667ex;" xmlns="http://www.w3.org/2000/svg" width="3.041ex" height="2.213ex" role="img" focusable="false" viewbox="0 -683 1344.3 978"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="M_j"><g data-mml-node="msub" data-latex="M_j"><g data-mml-node="mi" data-latex="M"><path data-c="1D440" d="M594 16C594 5 600 0 613 0L736 3C759 4 840 0 859 0C874 0 882 8 882 24C882 34 872 39 851 39C810 39 790 43 790 52C790 55 792 62 795 75L927 602C932 623 942 636 955 641C961 643 979 644 1008 644C1033 644 1044 645 1044 668C1044 678 1034 683 1013 683L882 683C853 683 853 681 840 662L484 108L408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660C204 649 215 644 236 644C256 644 297 647 297 631C297 629 296 623 293 613L167 110C158 72 135 49 100 42C75 39 42 43 42 16C42 5 48 0 60 0C78 0 141 3 159 3C178 3 242 0 261 0C276 0 283 8 283 24C283 33 276 38 261 39C219 39 198 52 198 78C198 82 199 89 202 100L332 621L415 26C418 9 424 0 434 0C442 0 450 7 459 20L848 627L712 82C706 60 696 47 681 42C675 40 656 39 625 39C604 39 594 37 594 16Z"/></g><g data-mml-node="mi" transform="translate(1003,-150) scale(0.707)" data-latex="j"><path data-c="1D457" d="M397 621C397 648 383 661 356 661C328 661 300 633 300 605C300 578 313 565 340 565C368 565 397 593 397 621M267 444C214 444 169 404 147 369C120 326 107 299 107 287C107 278 112 274 122 274C127 274 130 275 133 276C138 284 141 291 144 296C176 375 216 414 264 414C282 414 291 400 291 373C291 358 289 341 284 323L192-46C178-104 138-175 75-175C66-175 57-174 48-171C73-161 85-143 85-118C85-92 71-79 44-79C12-79-13-108-13-140C-13-184 30-205 77-205C120-205 160-190 197-160C232-131 254-94 265-51L356 311C359 324 361 337 361 349C361 404 322 444 267 444Z"/></g></g></g></g></svg></mjx-container> 为矩阵<mjx-container class="MathJax" jax="SVG" overflow="overflow"><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" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g></g></g></svg></mjx-container>的第<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="0.932ex" height="1.959ex" role="img" focusable="false" viewbox="0 -661 412 866"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="j"><g data-mml-node="mi" data-latex="j"><path data-c="1D457" d="M397 621C397 648 383 661 356 661C328 661 300 633 300 605C300 578 313 565 340 565C368 565 397 593 397 621M267 444C214 444 169 404 147 369C120 326 107 299 107 287C107 278 112 274 122 274C127 274 130 275 133 276C138 284 141 291 144 296C176 375 216 414 264 414C282 414 291 400 291 373C291 358 289 341 284 323L192-46C178-104 138-175 75-175C66-175 57-174 48-171C73-161 85-143 85-118C85-92 71-79 44-79C12-79-13-108-13-140C-13-184 30-205 77-205C120-205 160-190 197-160C232-131 254-94 265-51L356 311C359 324 361 337 361 349C361 404 322 444 267 444Z"/></g></g></g></svg></mjx-container>个余子式,<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.667ex;" xmlns="http://www.w3.org/2000/svg" width="7.142ex" height="2.36ex" role="img" focusable="false" viewbox="0 -748 3156.7 1043"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="λ_k(M_j)"><g data-mml-node="msub" data-latex="λ_k"><g data-mml-node="mi" data-latex="λ"><path data-c="1D706" d="M86-13C97-13 109-6 122 9L355 285C390 189 416 117 433 68C444 37 451 19 456 12C467-3 486-11 512-11L533-11C543-10 548-6 548 3C548 5 546 9 542 14C535 21 528 36 520 59L319 621C302 670 261 694 196 694C181 694 174 689 174 679C174 672 180 667 191 664C205 660 214 655 219 649C226 641 236 621 247 589L342 322L72 53C59 40 53 29 53 20C53 2 68-13 86-13Z"/></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)" data-latex="k"><path data-c="1D458" d="M409 353C409 327 423 314 450 314C485 314 508 345 508 379C508 418 476 445 437 445C392 445 344 415 291 356C250 311 217 282 190 269L291 679C289 688 287 694 274 694C242 694 166 685 154 684C139 682 132 675 132 660C132 650 141 645 159 645C178 645 204 646 204 632L59 43C56 32 55 25 55 21C55 0 66-11 87-11C104-11 117-3 124 12C129 21 147 92 179 226C231 221 286 196 286 146C286 131 279 101 279 91C279 34 316-11 373-11C431-11 470 41 490 145C490 154 485 159 475 159C466 159 460 152 457 138C435 59 408 19 375 19C357 19 348 33 348 61C348 77 360 131 360 147C360 204 314 239 221 253C244 269 270 292 298 322C326 352 346 371 359 382C386 404 412 415 435 415C445 415 453 413 459 409C432 404 409 379 409 353Z"/></g></g><g data-mml-node="mo" data-latex="(" transform="translate(1034.4,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="msub" data-latex="M_j" transform="translate(1423.4,0)"><g data-mml-node="mi" data-latex="M"><path data-c="1D440" d="M594 16C594 5 600 0 613 0L736 3C759 4 840 0 859 0C874 0 882 8 882 24C882 34 872 39 851 39C810 39 790 43 790 52C790 55 792 62 795 75L927 602C932 623 942 636 955 641C961 643 979 644 1008 644C1033 644 1044 645 1044 668C1044 678 1034 683 1013 683L882 683C853 683 853 681 840 662L484 108L408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660C204 649 215 644 236 644C256 644 297 647 297 631C297 629 296 623 293 613L167 110C158 72 135 49 100 42C75 39 42 43 42 16C42 5 48 0 60 0C78 0 141 3 159 3C178 3 242 0 261 0C276 0 283 8 283 24C283 33 276 38 261 39C219 39 198 52 198 78C198 82 199 89 202 100L332 621L415 26C418 9 424 0 434 0C442 0 450 7 459 20L848 627L712 82C706 60 696 47 681 42C675 40 656 39 625 39C604 39 594 37 594 16Z"/></g><g data-mml-node="mi" transform="translate(1003,-150) scale(0.707)" data-latex="j"><path data-c="1D457" d="M397 621C397 648 383 661 356 661C328 661 300 633 300 605C300 578 313 565 340 565C368 565 397 593 397 621M267 444C214 444 169 404 147 369C120 326 107 299 107 287C107 278 112 274 122 274C127 274 130 275 133 276C138 284 141 291 144 296C176 375 216 414 264 414C282 414 291 400 291 373C291 358 289 341 284 323L192-46C178-104 138-175 75-175C66-175 57-174 48-171C73-161 85-143 85-118C85-92 71-79 44-79C12-79-13-108-13-140C-13-184 30-205 77-205C120-205 160-190 197-160C232-131 254-94 265-51L356 311C359 324 361 337 361 349C361 404 322 444 267 444Z"/></g></g><g data-mml-node="mo" data-latex=")" transform="translate(2767.7,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></mjx-container>是该主子式的第<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" 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" data-latex="k"><g data-mml-node="mi" data-latex="k"><path data-c="1D458" d="M409 353C409 327 423 314 450 314C485 314 508 345 508 379C508 418 476 445 437 445C392 445 344 415 291 356C250 311 217 282 190 269L291 679C289 688 287 694 274 694C242 694 166 685 154 684C139 682 132 675 132 660C132 650 141 645 159 645C178 645 204 646 204 632L59 43C56 32 55 25 55 21C55 0 66-11 87-11C104-11 117-3 124 12C129 21 147 92 179 226C231 221 286 196 286 146C286 131 279 101 279 91C279 34 316-11 373-11C431-11 470 41 490 145C490 154 485 159 475 159C466 159 460 152 457 138C435 59 408 19 375 19C357 19 348 33 348 61C348 77 360 131 360 147C360 204 314 239 221 253C244 269 270 292 298 322C326 352 346 371 359 382C386 404 412 415 435 415C445 415 453 413 459 409C432 404 409 379 409 353Z"/></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="/go.html?u=aHR0cHM6Ly9hcnhpdi5vcmcvYWJzLzE5MDguMDM3OTU">Eigenvectors from Eigenvalues</a></p><p><a target="_blank" rel="external nofollow noopener noreferrer" href="/go.html?u=aHR0cHM6Ly9hcnhpdi5vcmcvYWJzLzE5MDcuMDI1MzQ">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 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