{"id":3048,"date":"2023-03-08T22:38:27","date_gmt":"2023-03-08T14:38:27","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3048"},"modified":"2025-02-26T11:02:47","modified_gmt":"2025-02-26T03:02:47","slug":"linear_algebra_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2023\/03\/08\/linear_algebra_mark\/","title":{"rendered":"\u7ebf\u6027\u4ee3\u6570\u6ce8\u8bb0"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u4eca\u5929\u7ec8\u4e8e\u4ece\u7f51\u6613\u79bb\u804c\u4e86, \u4ece\u7f51\u6613\u5927\u95e8\u79bb\u5f00\u65f6, \u771f\u7684\u662f\u611f\u6168\u4e07\u5343, \u671f\u5f85\u8fd8\u4f1a\u6709\u518d\u4f1a\u7684\u4e00\u5929\u53ed, \u6bd5\u7adf\u8fd9\u662f\u81ea\u5df1\u804c\u4e1a\u751f\u6daf\u7684\u8d77\u70b9~ \u5c71\u9ad8\u6c34\u957f, \u5e0c\u671b\u80fd\u548c\u670b\u53cb\u4eec\u6c5f\u6e56\u518d\u89c1~ \u56de\u5f52\u6b63\u9898, \u76ee\u524d\u8bfb\u5230\u4e86Shirley P, Ashikhmin M, Marschner S. Fundamentals of computer graphics[M]. AK Peters\/CRC Press, 2009.\u7684\u7b2c5\u7ae0Linear Algebra, \u7531\u4e8e\u81ea\u5df1\u9ad8\u8003\u540e\u4e5f\u5b66\u8fc7, \u6545\u672c\u6587\u53ea\u662f\u4f5c\u4e00\u4e9b\u8981\u70b9\u4e0a\u7684\u590d\u4e60, \u4e0d\u4f1a\u7eaa\u5f55\u5f97\u592a\u8be6\u7ec6~<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/ccjou.wordpress.com\/2014\/03\/05\/%E5%88%A9%E7%94%A8%E8%A1%8C%E5%88%97%E5%BC%8F%E6%B1%82%E7%9B%B4%E7%B7%9A%E3%80%81%E5%B9%B3%E9%9D%A2%E5%92%8C%E5%9C%93%E6%96%B9%E7%A8%8B%E5%BC%8F\/\">\u5229\u7528\u884c\u5217\u5f0f\u6c42\u76f4\u7dda\u3001\u5e73\u9762\u548c\u5713\u65b9\u7a0b\u5f0f<\/a><br \/>\n2. <a href=\"https:\/\/blog.csdn.net\/jiuzaizuotian2014\/article\/details\/111773918\">\u5217\u6b63\u4ea4\u77e9\u9635\u4e00\u5b9a\u662f\u884c\u6b63\u4ea4\u77e9\u9635\uff0c\u53cd\u4e4b\u4ea6\u7136<\/a><br \/>\n3. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/297827171\">\u77e9\u9635\u4e58\u79ef\u7684\u884c\u5217\u5f0f\u7b49\u4e8e\u884c\u5217\u5f0f\u7684\u4e58\u79ef |AB|=|A||B|<\/a><br \/>\n4. <a href=\"https:\/\/www.zhihu.com\/question\/38522020\">\u901a\u4fd7\u5730\u89e3\u91ca\u884c\u5217\u5f0f\u4e0e\u5176\u8f6c\u7f6e\u884c\u5217\u5f0f\u76f8\u7b49\u7684\u539f\u56e0\uff1f<\/a><br \/>\n5. <a href=\"https:\/\/baike.baidu.com\/item\/%E6%B7%B7%E5%90%88%E7%A7%AF\/10564182\">\u6df7\u5408\u79ef<\/a><br \/>\n6. <a href=\"https:\/\/math.stackexchange.com\/questions\/1603651\/volume-of-tetrahedron-using-cross-and-dot-product\">Volume of tetrahedron using cross and dot product<\/a><br \/>\n7. <a href=\"https:\/\/math.stackexchange.com\/questions\/1702271\/existence-and-uniqueness-of-the-eigen-decomposition-of-a-square-matrix\">Existence and uniqueness of the eigen decomposition of a square matrix<\/a><br \/>\n8. <a href=\"https:\/\/math.stackexchange.com\/questions\/235396\/eigenvalues-are-unique\">Eigenvalues are unique?<\/a><br \/>\n9. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/25793392\">\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u516c\u5f0f\u516b\u53d9<\/a><\/p>\n<p><strong>1. \u7279\u5f81\u503c\u4e0e\u77e9\u9635\u5bf9\u89d2\u5316<\/strong><\/p>\n<p>\u4e00\u4e2a\u65b9\u9635$\\mathbf{A}$\u603b\u6709\u7279\u5f81\u503c$\\lambda$\u4e0e\u7279\u5f81\u5411\u91cf$\\mathbf{a}$, \u800c\u7279\u5f81\u5411\u91cf$\\mathbf{a}$\u5219\u662f\u4e0e\u77e9\u9635$\\mathbf{A}$\u76f8\u4e58\u65f6\u65b9\u5411\u4e0d\u53d8\u7684\u975e\u96f6\u5411\u91cf, \u5373$$\\mathbf{A} \\mathbf{a} = \\lambda \\mathbf{a}.$$\u8fd9\u610f\u5473\u7740\u6211\u4eec\u62c9\u4f38\u6216\u538b\u7f29\u4e86$\\mathbf{a}$, \u4f46\u5b83\u7684\u65b9\u5411\u6ca1\u6709\u6539\u53d8. \u653e\u7f29\u56e0\u5b50$\\lambda$\u88ab\u79f0\u4e3a\u4e0e\u7279\u5f81\u5411\u91cf$\\mathbf{a}$\u76f8\u5173\u7684\u7279\u5f81\u503c. \u4e86\u89e3\u77e9\u9635\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u7684\u5404\u79cd\u5b9e\u9645\u5e94\u7528\u662f\u6709\u5e2e\u52a9\u7684. \u63a5\u4e0b\u6765\u5c06\u7b80\u5355\u4ecb\u7ecd\u4e00\u4e0b\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf, \u4ee5\u6df1\u5165\u4e86\u89e3\u51e0\u4f55\u53d8\u6362\u77e9\u9635, \u5e76\u4f5c\u4e3a\u8fc8\u5411\u4e0b\u4e00\u8282\u4e2d\u63cf\u8ff0\u7684\u5947\u5f02\u503c\u4e0e\u5947\u5f02\u5411\u91cf\u7684\u4e00\u6b65.<br \/>\n$\\\\$ \u82e5\u5047\u8bbe\u4e00\u4e2a\u65b9\u9635\u81f3\u5c11\u6709\u4e00\u4e2a\u7279\u5f81\u5411\u91cf, \u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5982\u4e0b\u65b9\u6cd5\u8fdb\u884c\u8ba1\u7b97. \u9996\u5148, \u6211\u4eec\u628a$\\mathbf{A} \\mathbf{a} = \\lambda \\mathbf{a}$\u4e24\u8fb9\u5199\u6210\u65b9\u9635\u4e0e\u5411\u91cf$\\mathbf{a}$\u7684\u4e58\u79ef:$$\\mathbf{A} \\mathbf{a} = \\lambda \\mathbf{I} \\mathbf{a},$$\u5176\u4e2d, $\\mathbf{I}$\u4e3a\u5355\u4f4d\u77e9\u9635. \u4e0a\u5f0f\u53ef\u4ee5\u88ab\u91cd\u5199\u4e3a$$\\mathbf{A} \\mathbf{a} &#8211; \\lambda \\mathbf{I} \\mathbf{a} = 0.$$\u7531\u4e8e\u77e9\u9635\u4e58\u6cd5\u6ee1\u8db3\u5206\u914d\u5f8b, \u6211\u4eec\u53ef\u5f97:$$(\\mathbf{A} &#8211; \\lambda \\mathbf{I})\\mathbf{a} = 0.$$\u77e9\u9635$(\\mathbf{A} &#8211; \\lambda \\mathbf{I})$\u4e3a\u5947\u5f02\u77e9\u9635\u5f53\u4e14\u4ec5\u5f53\u5176\u884c\u5217\u5f0f\u65b9\u4e3a\u96f6, \u4e14\u5176\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5747\u4e3a$\\mathbf{A}$\u4e2d\u7684\u5143\u7d20. \u4f8b\u5982, \u4e00\u4e2a$2 \\times 2$\u77e9\u9635\u7684\u7279\u5f81\u65b9\u7a0b\u4e3a$$\\begin{vmatrix}<br \/>\na_{11} &#8211; \\lambda &#038; a_{12}\\\\<br \/>\na_{21} &#038; a_{22} &#8211; \\lambda<br \/>\n\\end{vmatrix} = \\lambda^2 &#8211; (a_{11} + a_{22})\\lambda + (a_{11} a_{22} &#8211; a_{12} a_{21}) = 0.$$\u7531\u4e8e\u8fd9\u662f\u4e00\u4e2a\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b, \u6211\u4eec\u77e5\u9053$\\lambda$\u6709\u4e24\u4e2a\u503c, \u4f46\u4e0d\u4e00\u5b9a\u4e3a\u5b9e\u6570. \u5bf9\u4e8e$n \\times n$\u77e9\u9635\u7684\u7279\u5f81\u65b9\u7a0b, \u7531\u4e3aAbel-Ruffini\u5b9a\u7406\u53ef\u77e5, \u4e94\u6b21(\u542b) \u4ee5\u4e0a\u7684\u65b9\u7a0b\u6ca1\u6709\u6c42\u6839\u516c\u5f0f, \u8fd9\u91cc\u6240\u8c13\u7684\u6c42\u6839\u516c\u5f0f\u53ea\u6d89\u53ca\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\u4ee5\u53ca\u5f00\u4efb\u610f\u6b21\u6839, \u6545\u5bf9\u4e8e\u8f83\u5927\u7684\u77e9\u9635, \u6570\u503c\u65b9\u6cd5\u662f\u552f\u4e00\u7684\u9009\u62e9.<br \/>\n$\\\\$ \u5b9e\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u603b\u662f\u5b9e\u6570, \u4e14\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u662f\u76f8\u4e92\u6b63\u4ea4\u7684. \u5b9e\u5bf9\u79f0\u77e9\u9635\u53ef\u4ee5\u5199\u6210\u5bf9\u89d2\u7ebf\u5f62\u5f0f:$$\\mathbf{A} = \\mathbf{Q} \\mathbf{D} \\mathbf{Q}^T,$$\u5176\u4e2d, $\\mathbf{Q}$\u4e3a\u6b63\u4ea4\u77e9\u9635, $\\mathbf{D}$\u4e3a\u5bf9\u89d2\u77e9\u9635. \u6b63\u4ea4\u77e9\u9635$\\mathbf{Q}$\u7684\u6bcf\u4e00\u5217\u6784\u6210\u7684\u5411\u91cf\u5747\u4e3a\u5b9e\u5bf9\u79f0\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u5411\u91cf, \u5bf9\u89d2\u77e9\u9635$\\mathbf{D}$\u7684\u6bcf\u4e00\u4e2a\u5bf9\u89d2\u7ebf\u5143\u7d20\u5747\u4e3a\u5b9e\u5bf9\u79f0\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c. \u628a\u5b9e\u5bf9\u79f0\u77e9\u9635$\\mathbf{A}$\u5199\u4e3a\u4e0a\u8ff0\u5f62\u5f0f\u4e5f\u79f0\u4e3a\u5b9e\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u5206\u89e3, \u5b83\u628a\u5b9e\u5bf9\u79f0\u77e9\u9635$\\mathbf{A}$\u5206\u89e3\u6210\u51e0\u4e2a\u66f4\u7b80\u5355\u7684\u77e9\u9635\u7684\u4e58\u79ef, \u4ece\u800c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u5f97\u5230\u5b83\u7684\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c.<\/p>\n<p><strong>2. \u5947\u5f02\u503c\u5206\u89e3<\/strong><\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u5c06\u9488\u5bf9\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u5206\u89e3\u63a8\u5e7f\u5230\u975e\u5bf9\u79f0(\u751a\u81f3\u975e\u65b9\u9635) \u77e9\u9635, \u5373\u5947\u5f02\u503c\u5206\u89e3(SVD). \u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u5206\u89e3\u4e0e\u975e\u5bf9\u79f0\u77e9\u9635\u7684\u5947\u5f02\u503c\u5206\u89e3\u7684\u4e3b\u8981\u533a\u522b\u5728\u4e8e, \u5206\u89e3\u5f97\u5230\u7684\u5de6\u53f3\u6b63\u4ea4\u77e9\u9635\u5728\u5947\u5f02\u503c\u5206\u89e3\u4e2d\u4e0d\u8981\u6c42\u76f8\u540c: $\\mathbf{A} = \\mathbf{U} \\mathbf{S} \\mathbf{V}^T.$\u5176\u4e2d, $\\mathbf{U}$, $\\mathbf{V}$\u4e3a\u4e24\u4e2a\u53ef\u80fd\u4e0d\u540c\u7684\u6b63\u4ea4\u77e9\u9635, \u5176\u5217\u5206\u522b\u79f0\u4e3a\u77e9\u9635$\\mathbf{A}$\u7684\u5de6\u5947\u5f02\u5411\u91cf\u4e0e\u53f3\u5947\u5f02\u5411\u91cf; $\\mathbf{S}$\u4e3a\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635, \u5176\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u79f0\u4e3a\u77e9\u9635$\\mathbf{A}$\u7684\u5947\u5f02\u503c. \u5f53\u77e9\u9635$\\mathbf{A}$\u4e3a\u5bf9\u79f0\u77e9\u9635\u4e14\u5176\u7279\u5f81\u503c\u5747\u4e3a\u975e\u8d1f\u6570\u65f6, SVD\u4e0e\u7279\u5f81\u503c\u5206\u89e3\u5f97\u5230\u7684\u7ed3\u679c\u662f\u76f8\u540c\u7684.<br \/>\n$\\\\$ \u5947\u5f02\u503c\u4e0e\u7279\u5f81\u503c\u4e4b\u95f4\u8fd8\u6709\u53e6\u4e00\u79cd\u5173\u7cfb, \u501f\u6b64\u53ef\u4ee5\u7528\u4e8e\u8ba1\u7b97SVD(\u5c3d\u7ba1\u8fd9\u5e76\u975e\u5de5\u4e1a\u754c\u7684\u6807\u51c6\u505a\u6cd5). \u9996\u5148, \u6211\u4eec\u5b9a\u4e49\u77e9\u9635$\\mathbf{M} = \\mathbf{A} \\mathbf{A}^T$, \u5e76\u5047\u8bbe\u53ef\u4ee5\u5bf9\u77e9\u9635$\\mathbf{M}$\u8fdb\u884cSVD:$$\\mathbf{M} = \\mathbf{A} \\mathbf{A}^T = (\\mathbf{U} \\mathbf{S} \\mathbf{V}^T)(\\mathbf{U} \\mathbf{S} \\mathbf{V}^T)^T = \\mathbf{U} \\mathbf{S} (\\mathbf{V}^T \\mathbf{V}) \\mathbf{S} \\mathbf{U}^T = \\mathbf{U} \\mathbf{S}^2 \\mathbf{U}^T.$$\u663e\u7136, \u77e9\u9635$\\mathbf{M}$\u4e3a\u5bf9\u79f0\u77e9\u9635, $\\mathbf{U} \\mathbf{S}^2 \\mathbf{U}^T$\u4e3a\u5176\u7279\u5f81\u503c\u5206\u89e3, $\\mathbf{S}^2$\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u4e3a\u77e9\u9635$\\mathbf{M}$\u7684\u6240\u6709\u975e\u8d1f\u7279\u5f81\u503c. \u56e0\u6b64, \u6613\u77e5\u4e00\u4e2a\u77e9\u9635\u7684\u5947\u5f02\u503c\u4e3a\u8be5\u77e9\u9635\u4e0e\u8be5\u77e9\u9635\u7684\u8f6c\u7f6e\u7684\u4e58\u79ef\u7684\u7279\u5f81\u503c\u7684\u5e73\u65b9\u6839, \u5176\u5de6\u5947\u5f02\u5411\u91cf\u4e3a\u8be5\u4e58\u79ef\u77e9\u9635\u7684\u7279\u5f81\u5411\u91cf. \u7c7b\u4f3c\u5730, \u77e9\u9635$\\mathbf{A}$\u7684\u53f3\u5947\u5f02\u5411\u91cf\u4e3a\u4e58\u79ef\u77e9\u9635$\\mathbf{A}^T \\mathbf{A}$\u7684\u7279\u5f81\u5411\u91cf.<\/p>\n<p><strong>3. \u5e38\u89c1\u95ee\u9898<\/strong><\/p>\n<p><strong>\u95ee1:<\/strong> \u4e3a\u4ec0\u4e48\u77e9\u9635\u4e58\u6cd5\u4e0d\u5b9a\u4e49\u4e3a\u9010\u5143\u7d20\u4e58\u79ef?<br \/>\n$\\\\$ <strong>\u7b541:<\/strong> \u9010\u5143\u7d20\u4e58\u79ef\u786e\u5b9e\u662f\u4e00\u4e2a\u5b9a\u4e49\u77e9\u9635\u4e58\u6cd5\u7684\u597d\u65b9\u6cd5. \u7136\u800c, \u5b83\u5e76\u4e0d\u662f\u5f88\u5b9e\u7528. \u6bd5\u7adf, \u5927\u591a\u6570\u77e9\u9635\u90fd\u662f\u7528\u6765\u53d8\u6362\u5217\u5411\u91cf\u7684. \u4f8b\u5982, \u57283D\u7a7a\u95f4\u4e2d$$\\mathbf{b} = \\mathbf{M} \\mathbf{a},$$\u5176\u4e2d, $\\mathbf{a}$, $\\mathbf{b}$\u4e3a3\u7ef4\u5411\u91cf, $\\mathbf{M}$\u4e3a$3 \\times 3$\u77e9\u9635. \u4e3a\u4e86\u652f\u6301\u65cb\u8f6c\u7b49\u51e0\u4f55\u8fd0\u7b97, 3\u7ef4\u5411\u91cf$\\mathbf{b}$\u7684\u6bcf\u4e2a\u5143\u7d20\u5747\u4e3a3\u7ef4\u5411\u91cf$\\mathbf{a}$\u7684\u6240\u6709\u5143\u7d20\u7684\u7ec4\u5408, \u8fd9\u4fbf\u8981\u6c42\u6211\u4eec\u9700\u8981\u9010\u884c\u6216\u9010\u5217\u5730\u5904\u7406\u77e9\u9635$\\mathbf{M}$, \u8fd9\u4ea6\u4e3a\u77e9\u9635\u4e58\u6cd5\u5b9a\u4e49\u7684\u57fa\u7840. \u6b64\u5916, \u77e9\u9635\u4e0e\u5411\u91cf\u4e4b\u95f4\u7684\u4e58\u6cd5\u6ee1\u8db3\u7ed3\u5408\u5f8b:$$\\mathbf{M}_2(\\mathbf{M}_1 \\mathbf{a}) = (\\mathbf{M}_2 \\mathbf{M}_1)\\mathbf{a},$$\u8fd9\u610f\u5473\u7740\u77e9\u9635\u4e58\u6cd5\u5141\u8bb8\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u590d\u5408\u77e9\u9635$\\mathbf{C} = \\mathbf{M}_2 \\mathbf{M}_1$\u6765\u53d8\u6362\u4e00\u4e2a\u5411\u91cf.<\/p>\n<p><strong>\u95ee2:<\/strong> \u4ec0\u4e48\u60c5\u51b5\u4e0b\u77e9\u9635\u7684\u7279\u5f81\u503c\u4e0e\u5176\u5947\u5f02\u503c\u662f\u4e00\u6837\u7684, \u800c\u4ec0\u4e48\u60c5\u51b5\u4e0b\u77e9\u9635\u7684\u5947\u5f02\u503c\u4e3a\u5176\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u7684\u5e73\u65b9?<br \/>\n$\\\\$ <strong>\u7b542:<\/strong> \u82e5\u4e00\u4e2a\u5b9e\u77e9\u9635$\\mathbf{A}$\u662f\u5bf9\u79f0\u7684, \u5e76\u4e14\u5176\u7279\u5f81\u503c\u5747\u4e3a\u975e\u8d1f\u6570, \u5219\u5176\u7279\u5f81\u503c\u4e0e\u5947\u5f02\u503c\u662f\u76f8\u540c\u7684. \u800c\u82e5\u77e9\u9635$\\mathbf{A}$\u4e0d\u5bf9\u79f0, \u5219\u77e9\u9635$\\mathbf{M} = \\mathbf{A} \\mathbf{A}^T$\u662f\u5bf9\u79f0\u7684, \u4e14\u77e9\u9635$\\mathbf{M}$\u5177\u6709\u975e\u8d1f\u7684\u5b9e\u7279\u5f81\u503c. \u6b64\u5916, \u77e9\u9635$\\mathbf{A}$\u4e0e\u5176\u8f6c\u7f6e\u77e9\u9635$\\mathbf{A}^T$\u7684\u7279\u5f81\u503c\u662f\u76f8\u540c\u7684, \u5747\u4e3a\u77e9\u9635$\\mathbf{M}$\u7684\u5947\u5f02\u503c\/\u7279\u5f81\u503c\u7684\u5e73\u65b9\u6839.<\/p>\n<p><strong>4. \u76f8\u5173\u4e60\u9898<\/strong><\/p>\n<p><strong>4.1<\/strong> \u75282\u7ef4\u884c\u5217\u5f0f\u5199\u51fa\u4e00\u6761\u7ecf\u8fc7\u70b9$(x_0, y_0)$\u4e0e\u70b9$(x_1, y_1)$\u76842\u7ef4\u76f4\u7ebf\u7684\u9690\u5f0f\u65b9\u7a0b.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u5047\u8bbe\u76f4\u7ebf\u65b9\u7a0b\u4e3a$ax + by + c = 0$. \u56e0\u4e3a$(x_0, y_0)$\u4e0e$(x_1, y_1)$\u5728\u6b64\u76f4\u7ebf\u4e0a, \u672a\u77e5\u6570$a$, $b$\u4e0e$c$\u5fc5\u5b9a\u6ee1\u8db3$$ax_0 + by_0 + c = 0, \\\\ ax_1 + by_1 + c = 0.$$\u56e0\u4e3a\u5b58\u5728\u4e09\u4e2a\u672a\u77e5\u6570, \u5374\u53ea\u6709\u4e24\u4e2a\u65b9\u7a0b\u5f0f, \u4e0d\u59a8\u5c06\u76f4\u7ebf\u65b9\u7a0b\u5f0f\u52a0\u5165\u8054\u7acb\u65b9\u7a0b\u7ec4, \u5982\u4e0b:$$ax + by + c = 0, \\\\ ax_0 + by_0 + c = 0, \\\\ ax_1 + by_1 + c = 0,$$\u6216\u7528\u77e9\u9635\u5f62\u5f0f\u8868\u793a\u4e3a$$\\begin{bmatrix}<br \/>\nx &#038; y &#038; 1\\\\<br \/>\nx_0 &#038; y_0 &#038; 1\\\\<br \/>\nx_1 &#038; y_1 &#038; 1<br \/>\n\\end{bmatrix}\\begin{bmatrix}<br \/>\na \\\\<br \/>\nb \\\\<br \/>\nc<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\n0 \\\\<br \/>\n0 \\\\<br \/>\n0<br \/>\n\\end{bmatrix}.$$\u56e0\u4e3a$k(ax + by + c) = 0$, $k \\ne 0$, \u4ee3\u8868\u540c\u4e00\u6761\u76f4\u7ebf, \u65e0\u9650\u591a\u7ec4$(a, b, $$ c)$\u6ee1\u8db3\u8fd9\u4e2a\u9f50\u6b21\u65b9\u7a0b. \u7531\u6b64\u63a8\u8bba\u7cfb\u6570\u77e9\u9635\u4e0d\u53ef\u9006, \u884c\u5217\u5f0f\u5fc5\u5b9a\u4e3a0, \u4e8e\u662f\u5bfc\u51fa\u76f4\u7ebf\u65b9\u7a0b\u5f0f\u7684\u884c\u5217\u5f0f\u8868\u8fbe:$$\\begin{vmatrix}<br \/>\nx &#038; y &#038; 1\\\\<br \/>\nx_0 &#038; y_0 &#038; 1\\\\<br \/>\nx_1 &#038; y_1 &#038; 1<br \/>\n\\end{vmatrix} = 0.$$\u4ece\u51e0\u4f55\u9762\u89e3\u91ca, \u60f3\u60f3\u6211\u4eec\u5c06$XOY$\u5e73\u9762\u4e0a\u4e09\u4e2a\u5171\u7ebf\u70b9$(x, y)$, $(x_0, y_0)$\u4e0e$(x_1, y_1)$\u6cbf\u7740$z$\u8f74\u79fb\u52a8\u4e00\u5355\u4f4d, \u65b0\u7684\u7a7a\u95f4\u5750\u6807\u5373\u4e3a$(x, y, 1)$, $(x_0, y_0, 1)$\u4e0e$(x_1, y_1, 1)$. \u8fd9\u4e09\u4e2a\u5411\u91cf\u4f4d\u4e8e\u540c\u4e00\u5e73\u9762\u4e0a, \u56e0\u6b64\u6240\u5f20\u7684\u5e73\u884c\u516d\u9762\u4f53\u4f53\u79ef\u4e3a0, \u5373\u77e5\u4e09\u5411\u91cf\u5408\u5e76\u6210\u7684\u884c\u5217\u5f0f\u7b49\u4e8e0.<br \/>\n$\\\\$ \u7ee7\u7eed\u4f7f\u7528\u884c\u5217\u5f0f\u7b80\u5316\u76f4\u7ebf\u65b9\u7a0b\u5f0f,$$\\begin{vmatrix}<br \/>\nx &#038; y &#038; 1\\\\<br \/>\nx_0 &#038; y_0 &#038; 1\\\\<br \/>\nx_1 &#038; y_1 &#038; 1<br \/>\n\\end{vmatrix} = \\begin{vmatrix}<br \/>\nx &#038; y &#038; 1\\\\<br \/>\nx_0 &#038; y_0 &#038; 1\\\\<br \/>\nx_1 &#8211; x_0 &#038; y_1 &#8211; y_0 &#038; 0<br \/>\n\\end{vmatrix} = \\begin{vmatrix}<br \/>\nx &#038; y &#038; 1\\\\<br \/>\nx_0 &#8211; x &#038; y_0 &#8211; y &#038; 0\\\\<br \/>\nx_1 &#8211; x_0 &#038; y_1 &#8211; y_0 &#038; 0<br \/>\n\\end{vmatrix} \\\\ = \\begin{vmatrix}<br \/>\nx_0 &#8211; x &#038; y_0 &#8211; y\\\\<br \/>\nx_1 &#8211; x_0 &#038; y_1 &#8211; y_0<br \/>\n\\end{vmatrix} = (x_0 &#8211; x)(y_1 &#8211; y_0) &#8211; (x_1 &#8211; x_0)(y_0 &#8211; y) = 0,$$\u53ef\u5f97\u70b9\u659c\u5f0f$$y &#8211; y_0 = \\frac{y_1 &#8211; y_0}{x_1 &#8211; x_0}(x &#8211; x_0),$$\u5176\u4e2d, $\\frac{y_1 &#8211; y_0}{x_1 &#8211; x_0}$\u662f\u76f4\u7ebf\u7684\u659c\u7387.<\/p>\n<p><strong>4.2<\/strong> \u8bc1\u660e\u5982\u679c\u77e9\u9635\u7684\u5217\u5411\u91cf\u662f\u76f8\u4e92\u6b63\u4ea4\u7684, \u90a3\u4e48\u884c\u5411\u91cf\u4e5f\u662f\u76f8\u4e92\u6b63\u4ea4\u7684.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u82e5\u77e9\u9635$\\mathbf{A}$\u4e3a\u5217\u6b63\u4ea4\u77e9\u9635, \u5219\u77e9\u9635$\\mathbf{A}$\u7684\u6bcf\u4e2a\u5217\u5411\u91cf\u4e24\u4e24\u6b63\u4ea4, \u5373\u4efb\u610f\u4e24\u4e2a\u5217\u5411\u91cf\u5185\u79ef\u4e3a0, \u4efb\u610f\u4e00\u4e2a\u5217\u5411\u91cf\u4e0e\u81ea\u8eab\u7684\u5185\u79ef\u4e3a1, \u5219\u663e\u7136\u6709$\\mathbf{A}^T $$ \\mathbf{A} = \\mathbf{I}$. \u7531\u4e8e\u4e24\u4e24\u6b63\u4ea4, \u5219\u5176\u5217\u5411\u91cf\u7ec4\u7ebf\u6027\u65e0\u5173, \u77e9\u9635$\\mathbf{A}$\u6ee1\u79e9, \u77e9\u9635$\\mathbf{A}$\u5fc5\u7136\u53ef\u9006. \u4ece\u800c\u6211\u4eec\u6709$\\mathbf{A}^T = $$ \\mathbf{A}^{-1}$, \u5373$\\mathbf{A} \\mathbf{A}^T = \\mathbf{A} \\mathbf{A}^{-1} = \\mathbf{I}$, \u5373\u77e9\u9635$\\mathbf{A}$\u7684\u4efb\u610f\u4e24\u4e2a\u884c\u5411\u91cf\u5185\u79ef\u4e3a0, \u4efb\u610f\u4e00\u4e2a\u884c\u5411\u91cf\u4e0e\u81ea\u8eab\u7684\u5185\u79ef\u4e3a1. \u7efc\u4e0a\u6240\u8ff0, \u77e9\u9635$\\mathbf{A}$\u7684\u884c\u5411\u91cf\u4e5f\u662f\u76f8\u4e92\u6b63\u4ea4\u7684, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p><strong>4.3<\/strong> \u5bf9\u4e8e\u4e24\u4e2a\u65b9\u9635$\\mathbf{A}$, $\\mathbf{B}$, \u8bc1\u660e\u4e0b\u8ff0\u77e9\u9635\u884c\u5217\u5f0f\u7684\u6027\u8d28:<br \/>\n$\\\\$ (1) $|\\mathbf{A} \\mathbf{B}| = |\\mathbf{A}| |\\mathbf{B}|$;<br \/>\n$\\\\$ (2) $|\\mathbf{A}^{-1}| = \\frac{1}{|\\mathbf{A}|}$;<br \/>\n$\\\\$ (3) $|\\mathbf{A}^T| = |\\mathbf{A}|$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> (1) \u5148\u5217\u51fa\u4e0b\u8ff0\u8bc1\u660e\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5f15\u7406:<br \/>\n$\\\\$ <strong>Lemma 1<\/strong> $\\begin{vmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{0} \\\\<br \/>\n\\mathbf{C} &#038; \\mathbf{B}<br \/>\n\\end{vmatrix} = |\\mathbf{A}||\\mathbf{B}|$.<br \/>\n$\\\\$ <strong>Lemma 2<\/strong> $\\begin{vmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{C} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{B}<br \/>\n\\end{vmatrix} = |\\mathbf{A}||\\mathbf{B}|$.<br \/>\n$\\\\$ <strong>Lemma 3<\/strong> $\\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; \\mathbf{0} \\\\<br \/>\n\\mathbf{X} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{B} \\\\<br \/>\n\\mathbf{C} &#038; \\mathbf{D}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix} = \\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{B} \\\\<br \/>\n\\mathbf{C} &#038; \\mathbf{D}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix}$(\u5229\u7528\u4e86\u884c\u5217\u5f0f\u7684\u6027\u8d28, \u628a\u67d0\u884c\u7684\u67d0\u500d\u52a0\u5230\u53e6\u5916\u4e00\u884c, \u884c\u5217\u5f0f\u4e0d\u53d8).<br \/>\n$\\\\$ <strong>Lemma 4<\/strong> $\\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; \\mathbf{X} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{B} \\\\<br \/>\n\\mathbf{C} &#038; \\mathbf{D}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix} = \\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{B} \\\\<br \/>\n\\mathbf{C} &#038; \\mathbf{D}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix}$.<br \/>\n$\\\\$ \u7efc\u5408\u4e0a\u8ff0\u5f15\u7406\u53ef\u5f97:$$|\\mathbf{A}||\\mathbf{B}| = \\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{0} \\\\<br \/>\n\\mathbf{I} &#038; \\mathbf{B}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix}(Lemma\\ 1) \\\\ = \\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; -\\mathbf{A} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{0} \\\\<br \/>\n\\mathbf{I} &#038; \\mathbf{B}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix}(Lemma\\ 4) \\\\ = \\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; \\mathbf{I} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; -\\mathbf{A} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{0} \\\\<br \/>\n\\mathbf{I} &#038; \\mathbf{B}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix}(Lemma\\ 4) \\\\ = \\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; \\mathbf{I} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; \\mathbf{0} \\\\<br \/>\n-\\mathbf{I} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; -\\mathbf{A} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{I}<br \/>\n\\end{pmatrix}\\begin{pmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{0} \\\\<br \/>\n\\mathbf{I} &#038; \\mathbf{B}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix}(Lemma\\ 3) \\\\ = \\begin{vmatrix}<br \/>\n\\begin{pmatrix}<br \/>\n\\mathbf{I} &#038; \\mathbf{B} &#8211; \\mathbf{A} \\mathbf{B} \\\\<br \/>\n\\mathbf{0} &#038; \\mathbf{A} \\mathbf{B}<br \/>\n\\end{pmatrix}<br \/>\n\\end{vmatrix} = |\\mathbf{I}||\\mathbf{A} \\mathbf{B}|(&#8212;Lemma\\ 2) = |\\mathbf{A} \\mathbf{B}|.$$<\/p>\n<p>(2) \u7531(1)\u53ef\u5f97, $1 = |\\mathbf{I}| = |\\mathbf{A} \\mathbf{A}^{-1}| = |\\mathbf{A}| |\\mathbf{A}^{-1}|$, \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>(3) \u8003\u8651\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362$$\\mathbf{y} = \\mathbf{A} \\mathbf{x},$$\u5b83\u7684\u51e0\u4f55\u610f\u4e49\u662f\u628a\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5411\u91cf$\\mathbf{x}$\u901a\u8fc7\u7ebf\u6027\u53d8\u6362$\\mathbf{A}$\u53d8\u5230\u53e6\u5916\u4e00\u4e2a\u5411\u91cf$\\mathbf{y}$. $|\\mathbf{A}|$\u5927\u6982\u8868\u793a\u4e86\u4ece\u5411\u91cf$\\mathbf{x}$\u53d8\u6362\u5230\u5411\u91cf$\\mathbf{y}$\u7684\u4e00\u4e2a\u7f29\u653e\u56e0\u5b50.<br \/>\n$\\\\$ \u6839\u636eSVD, $\\mathbf{A} \\mathbf{x}$\u53ef\u4ee5\u5206\u89e3\u6210$$\\mathbf{A} \\mathbf{x} = \\mathbf{U} \\mathbf{S} \\mathbf{V}^T \\mathbf{x},$$\u5176\u4e2d, $\\mathbf{U}$, $\\mathbf{V}$\u4e3a\u6b63\u4ea4\u77e9\u9635, \u51e0\u4f55\u610f\u4e49\u662f\u65cb\u8f6c. \u4e5f\u5c31\u662f\u8bf4\u7ebf\u6027\u53d8\u6362$A$\u4f5c\u7528\u4e8e\u5411\u91cf$\\mathbf{x}$\u7b49\u4ef7\u4e8e\u5148\u5bf9\u5411\u91cf$\\mathbf{x}$\u65cb\u8f6c\u5f97\u5230$\\mathbf{V}^T \\mathbf{x}$, \u518d\u7f29\u653e\u5f97\u5230$\\mathbf{S} \\mathbf{V}^T \\mathbf{x}$, \u6700\u540e\u518d\u65cb\u8f6c\u5f97\u5230$\\mathbf{U} \\mathbf{S} \\mathbf{V}^T \\mathbf{x}$.<br \/>\n$\\\\$ \u540c\u6837\u7684\u9053\u7406, $\\mathbf{A}^T \\mathbf{x}$\u53ef\u4ee5\u5206\u89e3\u6210$$\\mathbf{A}^T \\mathbf{x} = \\mathbf{V} \\mathbf{S} \\mathbf{U}^T \\mathbf{x}.$$\u56e0\u4e3a\u5f71\u54cd\u7f29\u653e\u7684\u5bf9\u89d2\u77e9\u9635$S$\u6ca1\u6709\u53d8\u5316, \u6240\u4ee5$|\\mathbf{A}| = |\\mathbf{A}^T|$.<\/p>\n<p><strong>4.4<\/strong> \u8bc1\u660e\u5bf9\u89d2\u77e9\u9635\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u5373\u4e3a\u5176\u7279\u5f81\u503c.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u8bbe$n \\times n$\u5bf9\u89d2\u77e9\u9635$\\mathbf{A}$\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e3a$\\lambda_1, \\cdots, \\lambda_n$, \u5219\u5176\u7279\u5f81\u591a\u9879\u5f0f\u4e3a$$|\\mathbf{A} &#8211; \\lambda \\mathbf{I}| = (\\lambda &#8211; \\lambda_1) \\cdots (\\lambda &#8211; \\lambda_n),$$\u6545\u5bf9\u89d2\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u4e3a$\\lambda_1, \\cdots, \\lambda_n$, \u4ece\u800c\u547d\u9898\u5f97\u8bc1.<\/p>\n<p><strong>4.5<\/strong> \u8bc1\u660e\u5bf9\u4e8e\u65b9\u9635$\\mathbf{A}$, $\\mathbf{A} \\mathbf{A}^T$\u4e3a\u5bf9\u79f0\u77e9\u9635.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> $\\because (\\mathbf{A} \\mathbf{A}^T)^T = \\mathbf{A} \\mathbf{A}^T$, \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p><strong>4.6<\/strong> \u8bc1\u660e\u5bf9\u4e8e3\u4e2a3\u7ef4\u5411\u91cf$\\mathbf{a}$, $\\mathbf{b}$\u4e0e$\\mathbf{c}$, \u6709\u5982\u4e0b\u6052\u7b49\u5f0f\u6210\u7acb:$$|\\mathbf{a} \\mathbf{b} \\mathbf{c}| = (\\mathbf{a} \\times \\mathbf{b}) \\cdot \\mathbf{c}.$$<strong>\u8bc1:<\/strong> \u8bbe$$\\mathbf{a} = a_1 \\mathbf{i} + a_2 \\mathbf{j} + a_3 \\mathbf{k}, \\\\ \\mathbf{b} = b_1 \\mathbf{i} + b_2 \\mathbf{j} + b_3 \\mathbf{k}, \\\\ \\mathbf{c} = c_1 \\mathbf{i} + c_2 \\mathbf{j} + c_3 \\mathbf{k},$$\u5219\u6709$$(\\mathbf{a} \\times \\mathbf{b}) \\cdot \\mathbf{c} = \\begin{vmatrix}<br \/>\n\\mathbf{i} &#038; \\mathbf{j} &#038; \\mathbf{k}\\\\<br \/>\na_1 &#038; a_2 &#038; a_3\\\\<br \/>\nb_1 &#038; b_2 &#038; b_3<br \/>\n\\end{vmatrix} \\cdot (c_1 \\mathbf{i} + c_2 \\mathbf{j} + c_3 \\mathbf{k}) \\\\ = (\\mathbf{i} \\begin{vmatrix}<br \/>\na_2 &#038; a_3\\\\<br \/>\nb_2 &#038; b_3<br \/>\n\\end{vmatrix} &#8211; \\mathbf{j} \\begin{vmatrix}<br \/>\na_1 &#038; a_3\\\\<br \/>\nb_1 &#038; b_3<br \/>\n\\end{vmatrix} + \\mathbf{k} \\begin{vmatrix}<br \/>\na_1 &#038; a_2\\\\<br \/>\nb_1 &#038; b_2<br \/>\n\\end{vmatrix}) \\cdot (c_1 \\mathbf{i} + c_2 \\mathbf{j} + c_3 \\mathbf{k}) \\\\ = c_1 \\begin{vmatrix}<br \/>\na_2 &#038; a_3\\\\<br \/>\nb_2 &#038; b_3<br \/>\n\\end{vmatrix} &#8211; c_2  \\begin{vmatrix}<br \/>\na_1 &#038; a_3\\\\<br \/>\nb_1 &#038; b_3<br \/>\n\\end{vmatrix} + c_3 \\begin{vmatrix}<br \/>\na_1 &#038; a_2\\\\<br \/>\nb_1 &#038; b_2<br \/>\n\\end{vmatrix} = \\begin{vmatrix}<br \/>\na_1 &#038; a_2 &#038; a_3\\\\<br \/>\nb_1 &#038; b_2 &#038; b_3\\\\<br \/>\nc_1 &#038; c_2 &#038; c_3<br \/>\n\\end{vmatrix}.$$\u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p><strong>4.7<\/strong> \u8bc1\u660e\u8fb9\u5411\u91cf\u4e3a$\\mathbf{a}$, $\\mathbf{b}$\u4e0e$\\mathbf{c}$\u7684\u56db\u9762\u4f53\u4f53\u79ef\u4e3a$V = \\frac{1}{6} |\\mathbf{a} \\mathbf{b} \\mathbf{c}|$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u6ce8\u610f\u5230$$V = \\frac{1}{3}Bh = \\frac{1}{6} ||\\mathbf{a} \\times \\mathbf{b}|| \\cdot h,$$\u5176\u4e2d, $h = ||c|| \\cdot |cos \\theta|$, $\\theta$\u4e3a\u5411\u91cf$\\mathbf{a} \\times \\mathbf{b}$\u4e0e\u8fb9\u5411\u91cf$\\mathbf{c}$\u4e4b\u95f4\u7684\u5939\u89d2.<br \/>\n$\\\\$ \u53c8\u7531\u4e60\u98984.5\u77e5, $$|\\mathbf{a} \\mathbf{b} \\mathbf{c}| = (\\mathbf{a} \\times \\mathbf{b}) \\cdot \\mathbf{c} = ||\\mathbf{a} \\times \\mathbf{b}|| \\cdot ||c|| \\cdot |cos \\theta|,$$\u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n<p><strong>4.8<\/strong> \u8bc1\u660e\u82e5\u77e9\u9635$\\mathbf{A}$\u6709\u7279\u5f81\u503c\u5206\u89e3$\\mathbf{A} = \\mathbf{Q} \\mathbf{D} \\mathbf{Q}^T$, $\\mathbf{v}$\u4e3a\u6b63\u4ea4\u77e9\u9635$\\mathbf{Q}$\u7684\u7b2c$i$\u884c\u5143\u7d20\u6784\u6210\u7684\u5411\u91cf, $\\lambda$\u4e3a\u5bf9\u89d2\u77e9\u9635$\\mathbf{D}$\u5bf9\u89d2\u7ebf\u4e0a\u7684\u7b2c$i$\u4e2a\u5143\u7d20, \u5219$\\mathbf{v}$\u4e3a\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u5411\u91cf, \u5bf9\u5e94\u7684\u7279\u5f81\u503c\u4e3a$\\lambda$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4ee4$\\mathbf{e}_i$\u4e3a\u7b2c$i$\u4e2a\u5143\u7d20\u4e3a1, \u5176\u4f59\u5143\u7d20\u4e3a0\u7684\u5217\u5411\u91cf, s.t. $\\mathbf{v} = \\mathbf{Q} \\mathbf{e}_i$, \u5219$$\\mathbf{A} \\mathbf{v} = \\mathbf{A}(\\mathbf{Q} \\mathbf{e}_i) = \\mathbf{Q} \\mathbf{D} \\mathbf{Q}^T (\\mathbf{Q} \\mathbf{e}_i) = \\mathbf{Q}(\\mathbf{D} \\mathbf{e}_i) = \\mathbf{Q} d_{ii} \\mathbf{e}_i = d_{ii} (\\mathbf{Q} \\mathbf{e}_i),$$\u4ece\u800c$\\mathbf{v} = \\mathbf{Q} \\mathbf{e}_i$\u4e3a\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u5411\u91cf, \u5bf9\u5e94\u7684\u7279\u5f81\u503c\u4e3a$\\lambda$.<\/p>\n<p><strong>4.9<\/strong> \u8bc1\u660e\u82e5\u77e9\u9635$\\mathbf{A}$\u6709\u7279\u5f81\u503c\u5206\u89e3$\\mathbf{A} = \\mathbf{Q} \\mathbf{D} \\mathbf{Q}^T$, \u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u90fd\u662f\u4e0d\u540c\u7684, $\\mathbf{v}$\u4e3a\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u4e3a$\\lambda$\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf, \u5219$\\exists i$, s.t. \u7279\u5f81\u5411\u91cf$\\mathbf{v}$\u4e3a\u6b63\u4ea4\u77e9\u9635$\\mathbf{Q}$\u7684\u7b2c$i$\u884c\u5143\u7d20\u6784\u6210\u7684\u5411\u91cf, \u5bf9\u5e94\u7684\u7279\u5f81\u503c$\\lambda$\u4e3a\u5bf9\u89d2\u77e9\u9635$\\mathbf{D}$\u5bf9\u89d2\u7ebf\u4e0a\u7684\u7b2c$i$\u4e2a\u5143\u7d20.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u95ee\u9898\u5373\u8981\u8bc1: \u82e5\u77e9\u9635$\\mathbf{A}$\u6709\u7279\u5f81\u503c\u5206\u89e3$\\mathbf{A} = \\mathbf{Q} \\mathbf{D} \\mathbf{Q}^T$, \u4e14\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u90fd\u662f\u4e0d\u540c\u7684, \u5219\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u5206\u89e3\u662f\u552f\u4e00\u7684. \u8fd9\u610f\u5473\u7740\u5982\u4e0b\u4e24\u70b9\u4e8b\u5b9e\u6210\u7acb:<br \/>\n$\\\\$ (1) \u82e5\u77e9\u9635$\\mathbf{A}$\u6709\u53e6\u5916\u4e00\u4e2a\u7279\u5f81\u503c\u5206\u89e3$\\mathbf{A} = \\mathbf{E} \\mathbf{F} \\mathbf{E}^{-1}$, \u5219\u82e5\u4e0d\u8003\u8651\u5bf9\u89d2\u7ebf\u5143\u7d20\u7684\u987a\u5e8f, \u5bf9\u89d2\u77e9\u9635$\\mathbf{F}$\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e0e\u5bf9\u89d2\u77e9\u9635$\\mathbf{D}$\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u76f8\u540c.<br \/>\n$\\\\$ (2) \u82e5$\\mathbf{F} = \\mathbf{D}$, \u5219$\\mathbf{E} = \\mathbf{Q} \\mathbf{P}$, \u5176\u4e2d, $\\mathbf{P}$\u4e3a\u4e00\u4e2a\u53ef\u9006\u7684\u5bf9\u89d2\u77e9\u9635\u77e9\u9635(i.e. \u5bf9\u89d2\u5143\u7d20\u5747\u975e0).<br \/>\n$\\\\$ \u9996\u5148\u8bc1\u660e(1), \u7531\u4e8e$n \\times n$\u7684\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u4e3a\u7279\u5f81\u65b9\u7a0b$det(\\mathbf{A} &#8211; \\lambda \\mathbf{I}) $$ = 0$\u7684\u6839, \u4e14\u8be5\u7279\u5f81\u65b9\u7a0b\u4e3a\u4e00\u4e2a\u4e00\u5143$n$\u6b21\u65b9\u7a0b, \u7531\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\u53ef\u77e5, \u8be5\u7279\u5f81\u65b9\u7a0b\u6700\u591a\u53ea\u6709$n$\u4e2a\u6839, \u5373\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u6700\u591a\u53ea\u6709$n$\u4e2a. \u5728\u5bf9\u89d2\u77e9\u9635$\\mathbf{D}$\u4e3a\u4e00\u4e2a$n \\times n$\u7684\u77e9\u9635\u7684\u524d\u63d0\u4e0b, \u5bf9\u89d2\u77e9\u9635$\\mathbf{D}$\u7684\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5df2\u5305\u542b\u4e86$n$\u4e2a\u7279\u5f81\u503c(\u4e60\u98984.8), \u5bf9\u89d2\u77e9\u9635$\\mathbf{D}$\u7684\u5bf9\u89d2\u7ebf\u4e0a\u4e0d\u53ef\u80fd\u51fa\u73b0\u7b2c$n + 1$\u4e2a\u7279\u5f81\u503c, \u6545(1)\u5f97\u8bc1.<br \/>\n$\\\\$ \u518d\u8bc1\u660e(2), \u7531\u4e8e\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u503c\u90fd\u662f\u4e0d\u540c\u7684, \u5219\u6240\u6709\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u5747\u4e3a1\u7ef4\u7ebf\u6027\u7a7a\u95f4. \u53c8\u6b63\u4ea4\u77e9\u9635$\\mathbf{Q}$\u4e0e\u6b63\u4ea4\u77e9\u9635$\\mathbf{E}$\u7684\u7b2c$i$\u5217\u6784\u6210\u7684\u5411\u91cf\u5bf9\u5e94\u4e8e\u540c\u4e00\u4e2a\u7279\u5f81\u503c, \u6545\u5b83\u4eec\u4e00\u5b9a\u662f\u5e73\u884c\u7684, \u4ece\u800c\u53ef\u4ee5\u8bf1\u5bfc\u51fa\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635$\\mathbf{P}$, s.t. $\\mathbf{E} = \\mathbf{Q} \\mathbf{P}$.<br \/>\n$\\\\$ \u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p><strong>4.10<\/strong> \u7ed9\u5b9a\u4e00\u4e2a2\u7ef4\u4e09\u89d2\u5f62$\\triangle ABC$\u7684\u4e09\u4e2a\u9876\u70b9\u7684$(x, y)$\u5750\u6807($A(x_0, y_0)$, $B(x_1,<br \/>\n$$ y_1)$\u4e0e$C(x_2, y_2)$), \u8bc1\u660e\u5176\u9762\u79ef\u4e3a$$S_\\triangle = \\frac{1}{2} \\begin{vmatrix}<br \/>\nx_0 &#038; x_1 &#038; x_2\\\\<br \/>\ny_0 &#038; y_1 &#038; y_2\\\\<br \/>\n1 &#038; 1 &#038; 1<br \/>\n\\end{vmatrix}.$$<strong>\u8bc1:<\/strong> \u6211\u4eec\u91c7\u7528\u6784\u9020\u5df2\u77e5\u56fe\u5f62\u9762\u79ef\u7684\u65b9\u6cd5\u6765\u6c42\u89e3\u672a\u77e5\u56fe\u5f62\u9762\u79ef\u7684\u65b9\u6cd5, \u6240\u4ee5\u6211\u4eec\u5728\u5750\u6807\u7cfb\u91cc\u5bf9$\\triangle ABC$\u4e09\u4e2a\u9876\u70b9\u4f5c$x$\u8f74\u548c$y$\u8f74\u7684\u5782\u7ebf, \u4e8e\u662f$\\triangle ABC$\u7684\u9762\u79ef\u53ef\u4ee5\u770b\u6210\u662f\u4e00\u4e2a\u77e9\u5f62\u9762\u79ef\u51cf\u53bb\u4e09\u4e2a\u5468\u56f4\u7684\u5c0f\u4e09\u89d2\u5f62\u9762\u79ef$S_1$, $S_2$, $S_3$\u7684\u5dee, \u5373$$S_\\triangle = S &#8211; S_1 &#8211; S_2 &#8211; S_3,$$\u5176\u4e2d, $S$\u4e3a$\\triangle ABC$\u7684\u5916\u63a5\u77e9\u5f62\u7684\u9762\u79ef. \u77e9\u5f62\u9762\u79ef\u4f7f\u7528\u5e95\u4e58\u9ad8, \u4e09\u89d2\u5f62\u9762\u79ef\u4f7f\u7528\u5e95\u4e58\u9ad8\u7684\u4e00\u534a, \u53ef\u5f97:$$S = |x_2 &#8211; x_0| \\cdot |y_2 &#8211; y_1|, \\\\ S_1 = \\frac{1}{2} |x_2 &#8211; x_0| \\cdot |y_2 &#8211; y_0|, \\\\ S_2 = \\frac{1}{2} |x_1 &#8211; x_0| \\cdot |y_1 &#8211; y_0|, \\\\ S_3 = \\frac{1}{2} |x_2 &#8211; x_1| \\cdot |y_2 &#8211; y_1|,$$\u4ee3\u5165\u5316\u7b80\u53ef\u4ee5\u5f97\u5230:$$S_\\triangle = \\frac{1}{2} |(x_1 &#8211; x_0)(y_2 &#8211; y_0) &#8211; (x_2 &#8211; x_0)(y_1 &#8211; y_0)|,$$\u5199\u6210\u4e09\u9636\u884c\u5217\u5f0f\u7684\u5f62\u5f0f\u5c31\u662f:$$S_\\triangle = \\frac{1}{2} \\begin{vmatrix}<br \/>\nx_0 &#038; x_1 &#038; x_2\\\\<br \/>\ny_0 &#038; y_1 &#038; y_2\\\\<br \/>\n1 &#038; 1 &#038; 1<br \/>\n\\end{vmatrix}.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u5929\u7ec8\u4e8e\u4ece\u7f51\u6613\u79bb\u804c\u4e86, \u4ece\u7f51\u6613\u5927\u95e8\u79bb\u5f00\u65f6, \u771f\u7684\u662f\u611f\u6168\u4e07\u5343, \u671f\u5f85\u8fd8\u4f1a\u6709\u518d\u4f1a\u7684\u4e00\u5929\u53ed, \u6bd5\u7adf\u8fd9\u662f\u81ea\u5df1\u804c\u4e1a\u751f\u6daf\u7684 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2023\/03\/08\/linear_algebra_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u7ebf\u6027\u4ee3\u6570\u6ce8\u8bb0<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[27],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3048"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=3048"}],"version-history":[{"count":31,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3048\/revisions"}],"predecessor-version":[{"id":3605,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3048\/revisions\/3605"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3048"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3048"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3048"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}