{"id":2098,"date":"2023-04-01T22:10:30","date_gmt":"2023-04-01T14:10:30","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2098"},"modified":"2023-04-06T13:57:01","modified_gmt":"2023-04-06T05:57:01","slug":"detailed-calculation-process-of-matrix-eigenvalues-and-eigenvectors","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/01\/detailed-calculation-process-of-matrix-eigenvalues-and-eigenvectors\/","title":{"rendered":"\u77e9\u9635\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u8be6\u7ec6\u8ba1\u7b97\u8fc7\u7a0b"},"content":{"rendered":"

\u77e9\u9635\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u5b9a\u4e49<\/h2>\n

A<\/code>\u4e3an\u9636\u77e9\u9635\uff0c\u82e5\u6570\\lambda<\/code>\u548cn\u7ef4\u975e0\u5217\u5411\u91cfx<\/code>\u6ee1\u8db3Ax=\u03bbx<\/code>\uff0c\u90a3\u4e48\u6570\\lambda<\/code>\u79f0\u4e3aA<\/code>\u7684\u7279\u5f81\u503c\uff0cx<\/code>\u79f0\u4e3aA<\/code>\u7684\u5bf9\u5e94\u4e8e\u7279\u5f81\u503c\\lambda<\/code>\u7684\u7279\u5f81\u5411\u91cf\u3002\u5f0fAx=\u03bbx<\/code>\u4e5f\u53ef\u5199\u6210(A-\u03bbE)x=0<\/code>\uff0c\u5e76\u4e14|\u03bbE-A|<\/code>\u53eb\u505a$A$\u7684\u7279\u5f81\u591a\u9879\u5f0f\u3002\u5f53\u7279\u5f81\u591a\u9879\u5f0f\u7b49\u4e8e0\u7684\u65f6\u5019\uff0c\u79f0\u4e3a$A$\u7684\u7279\u5f81\u65b9\u7a0b\uff0c\u7279\u5f81\u65b9\u7a0b\u662f\u4e00\u4e2a\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u6c42\u89e3\u7279\u5f81\u503c\u7684\u8fc7\u7a0b\u5176\u5b9e\u5c31\u662f\u6c42\u89e3\u7279\u5f81\u65b9\u7a0b\u7684\u89e3\u3002<\/p>\n

<\/p>\n

Ax = \\lambda x \\implies Ax = \\lambda Ex \\implies (\\lambda E - A)x = 0<\/code><\/pre>\n
|\\lambda E - A| =\n\\begin {vmatrix}\n  \\lambda - a_{11} & -a_{12} & \\cdots & -a_{1n} \\\\\n  -a_{21} & \\lambda - a_{22} & \\cdots & -a_{2n} \\\\\n  \\vdots & \\vdots & \\ddots & \\vdots \\\\\n  -a_{n1} & -a_{n2} & \\cdots & \\lambda - a_{nn}\n\\end {vmatrix}\n= 0<\/code><\/pre>\n
\u7279\u5f81\u503c\u53ca\u7279\u5f81\u5411\u91cf\u8ba1\u7b97<\/h2>\n
\u82e5\u5df2\u77e5<\/p>\n
A =\n\\begin {pmatrix}\n  4 & 2 & -5 \\\\\n  6 & 4 & -9 \\\\\n  5 & 3 & -7\n\\end {pmatrix}<\/code><\/pre>\n
\u8ba1\u7b97\uff1aA\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf<\/p>\n
|\\lambda E - A| =\n\\begin {vmatrix}\n  \\lambda - 4 & -2 & 5 \\\\\n  -6 & \\lambda - 4 & 9 \\\\\n  -5 & -3 & \\lambda + 7\n\\end {vmatrix}\n= 0<\/code><\/pre>\n
\u8ba1\u7b97\u884c\u5217\u5f0f\u5f97\uff1a<\/p>\n
(\\lambda-4)(\\lambda-4)(\\lambda+7) + (-2)\\times9\\times(-5) + 5\\times(-6)\\times(-3) - \\\\\n(5\\times(\\lambda-4)\\times(-5) + (-2)\\times(-6)\\times(\\lambda+7) + (\\lambda-4)\\times9\\times(-3)) = 0<\/code><\/pre>\n
\u5316\u7b80\u5f97\uff1a<\/p>\n
\\lambda^2 * (\\lambda-1) = 0<\/code><\/pre>\n
\u5f97\u5230\u7279\u5f81\u503c\uff1a<\/p>\n
\\lambda_1 = 1, \\lambda_2 = \\lambda_3 = 0<\/code><\/pre>\n
\u5f53 \\lambda_1 = 1 \\implies (E - A)x = 0<\/code><\/p>\n
$E = \\bigl( \\begin{smallmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{smallmatrix} \\bigr)$ \u4ece\u800c $E - A = \\bigl( \\begin{smallmatrix} -3 & -2 & 5 \\\\ -6 & -3 & 9 \\\\ -5 & -3 & 8 \\end{smallmatrix} \\bigr)$ \u5316\u7b80\u5f97\uff1a$\\bigl( \\begin{smallmatrix} 0 & 1 & -1 \\\\ 1 & 0 & -1 \\\\ 0 & 0 & 0 \\end{smallmatrix} \\bigr)$<\/code><\/pre>\n
(E - A)x =\n\\begin {pmatrix}\n  0 & 1 & -1 \\\\\n  1 & 0 & -1 \\\\\n  0 & 0 & 0\n\\end {pmatrix}\n\\begin {pmatrix}\n  x_1 \\\\\n  x_2 \\\\\n  x_3\n\\end {pmatrix}\n= 0\n\\implies\n\\left \\\\{\n\\begin{array} {l}\nx_1 - x_3 = 0 \\\\\nx_2 - x_3 = 0\n\\end{array}\n\\right.<\/code><\/pre>\n
\u4ee4x_1 = 1<\/code>\uff0c\u5f97\u5230\u7279\u5f81\u77e9\u9635\uff1a<\/p>\n
\\xi_1 =\n\\begin {pmatrix}\n  1 \\\\\n  1 \\\\\n  1\n\\end {pmatrix}<\/code><\/pre>\n
\u540c\u7406\uff0c\u5f53\\lambda_2 = \\lambda_3 = 0<\/code>\u5f97\uff1a<\/p>\n
(E - A)x =\n\\begin {pmatrix}\n  -2 & 0 & 1 \\\\\n  0 & -2 & 3 \\\\\n  0 & 0 & 0\n\\end {pmatrix}\n\\begin {pmatrix}\n  x_1 \\\\\n  x_2 \\\\\n  x_3\n\\end {pmatrix}\n= 0\n\\implies\n\\left \\\\{\n\\begin{array} {l}\n-2x_1 + x_3 = 0 \\\\\n-2x_2 + 3x_3 = 0\n\\end{array}\n\\right.<\/code><\/pre>\n
\u4ee4x_1 = 1<\/code>\uff0c\u5f97\u5230\u7279\u5f81\u77e9\u9635\uff1a<\/p>\n
\\xi_2 = \\xi_3 =\n\\begin {pmatrix}\n  1 \\\\\n  3 \\\\\n  2\n\\end {pmatrix}<\/code><\/pre>\n
Python\u4e2d\u8ba1\u7b97\u7279\u5f81\u503c\u53ca\u7279\u5f81\u5411\u91cf<\/h2>\n
# -*- coding: utf-8 -*-\nimport numpy as np\n\na = np.array([[4, 2, -5], [6, 4, -9], [5, 3, -7]])\nprint(a)\n# x = np.linalg.eigvals(a)\n# print(x)\nw, v = np.linalg.eig(a)  # w\u662f\u7279\u5f81\u503c, v\u4e3a\u7279\u5f81\u5411\u91cf\nprint(w)\nprint(v)<\/code><\/pre>\n
\u8f93\u51fa\u7ed3\u679c\uff1a<\/p>\n
[[ 4  2 -5]\n [ 6  4 -9]\n [ 5  3 -7]]\n[ 1.00000000e+00  4.02000914e-08 -4.02000921e-08]\n[[ 0.57735027 -0.26726123  0.26726125]\n [ 0.57735027 -0.80178373  0.80178372]\n [ 0.57735027 -0.53452248  0.53452249]]<\/code><\/pre>\n
\n
\u6ce8\uff1anp.linalg\u5bf9\u77e9\u9635\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u8fdb\u884c\u4e86\u5f52\u4e00\u5316\u5904\u7406<\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":"
\u77e9\u9635\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u5b9a\u4e49 A\u4e3an\u9636\u77e9\u9635\uff0c\u82e5\u6570\\lambda\u548cn\u7ef4\u975e0\u5217\u5411\u91cfx\u6ee1\u8db3Ax=\u03bbx\uff0c\u90a3\u4e48\u6570\\lambd […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[527],"tags":[530],"class_list":["post-2098","post","type-post","status-publish","format-standard","hentry","category-ai","tag-530"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2098","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2098"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2098\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2098"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2098"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2098"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}