{"id":337,"date":"2025-09-17T00:10:27","date_gmt":"2025-09-16T16:10:27","guid":{"rendered":"https:\/\/blog.hcha.site\/?p=337"},"modified":"2026-03-07T14:10:56","modified_gmt":"2026-03-07T06:10:56","slug":"%e7%9f%a9%e9%98%b5%e4%b9%98%e6%b3%95%ef%bc%9a%e4%bb%8e%e5%98%89%e8%b1%aa%e7%9a%84%e4%b8%8b%e5%8d%88%e5%88%b0%e4%ba%ba%e5%b7%a5%e6%99%ba%e8%83%bd","status":"publish","type":"post","link":"https:\/\/blog.hcha.site\/?p=337","title":{"rendered":"\u77e9\u9635\u4e58\u6cd5\uff1a\u4ece\u5609\u8c6a\u7684\u4e0b\u5348\u5230\u4eba\u5de5\u667a\u80fd"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u8bb8\u591a\u5e74\u4ee5\u540e\uff0c\u5b66\u4e86\u77e9\u9635\u4e58\u6cd5\uff0c\u5609\u8c6a\u5c06\u4f1a\u56de\u60f3\u8d77\uff0c\u4ed6\u5c0f\u5b66\u8001\u5e08\u8b66\u544a\u4ed6\u4e58\u6cd5\u4e0d\u80fd\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u4ea4\u6362\u7684\u90a3\u4e2a\u4e0b\u5348\u3002<\/p>\n\n\n\n<p class=\"has-text-align-right\">\u2014\u2014\u300a\u767e\u5e74\u6570\u72ec\u300b<\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>\u5f15\u8a00\uff1a\u91cd\u5851\u201c\u4e58\u6cd5\u201d\u4e4b\u610f\u4e49<\/strong><\/h2>\n\n\n\n<p>\u201c\u4e58\u6cd5\u4ea4\u6362\u5f8b\u201d\uff0c\u8fd9\u662f\u4e00\u4e2a\u5982\u540c\u201c1+1=2\u201d\u4e00\u6837\uff0c\u81ea\u6211\u4eec\u9996\u6b21\u8e0f\u5165\u6570\u5b66\u6bbf\u5802\u4fbf\u88ab\u5949\u4e3a\u572d\u81ec\u7684\u516c\u7406\u3002\u5b83\u5982\u540c\u4e00\u5757\u575a\u5b9e\u7684\u57fa\u77f3\uff0c\u652f\u6491\u7740\u6211\u4eec\u6574\u4e2a\u7b97\u672f\u4e0e\u521d\u7b49\u4ee3\u6570\u7684\u5927\u53a6\u3002\u5bf9\u4e8e\u90a3\u4e2a\u5c0f\u5b66\u4e0b\u5348\u7684\u5609\u8c6a\u6765\u8bf4\uff0c<code>3 x 5<\/code> \u4e0e <code>5 x 3<\/code> \u5f97\u5230\u76f8\u540c\u7684\u7ed3\u679c\u662f\u5929\u7ecf\u5730\u4e49\u7684\uff0c\u5b83\u53cd\u6620\u4e86\u4e00\u79cd\u6df1\u523b\u7684\u5bf9\u79f0\u6027\u4e0e\u79e9\u5e8f\uff0c\u6211\u4eec\u5bf9\u4e16\u754c\u7684\u6734\u7d20\u8ba4\u77e5\u4e5f\u7531\u6b64\u5efa\u7acb\u3002\u7136\u800c\uff0c\u6570\u5b66\u7684\u5b8f\u4f1f\u7248\u56fe\u8fdc\u4e0d\u6b62\u4e8e\u6211\u4eec\u6700\u521d\u77a5\u89c1\u7684\u7530\u56ed\u98ce\u5149\u3002\u5728\u5b83\u7684\u6df1\u9083\u8179\u5730\uff0c\u5b58\u5728\u7740\u622a\u7136\u4e0d\u540c\u7684\u6cd5\u5219\u4e0e\u5927\u9646\uff0c\u800c\u77e9\u9635\u4e58\u6cd5\uff0c\u6b63\u662f\u6211\u4eec\u901a\u5f80\u8fd9\u7247\u65b0\u4e16\u754c\u7684\u7b2c\u4e00\u4e2a\uff0c\u4e5f\u662f\u6700\u5177\u98a0\u8986\u6027\u7684\u5173\u53e3\u3002<\/p>\n\n\n\n<p>\u5f53\u5609\u8c6a\u6700\u7ec8\u5728\u9ad8\u7b49\u6570\u5b66\u4e2d\u4e0e\u77e9\u9635\u4e58\u6cd5\u76f8\u9047\u65f6\uff0c\u4ed6\u6240\u9762\u4e34\u7684\u51b2\u51fb\u662f\u6839\u672c\u6027\u7684\u3002\u4ed6\u7684\u8001\u5e08\u4f1a\u544a\u8bc9\u4ed6\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u77e9\u9635 <code>A<\/code> \u4e58\u4ee5\u77e9\u9635 <code>B<\/code>\uff0c\u5e76\u4e0d\u7b49\u4e8e\u77e9\u9635 <code>B<\/code> \u4e58\u4ee5\u77e9\u9635 <code>A<\/code>\u3002\u8fd9\u4e0d\u4ec5\u4ec5\u662f\u4e00\u4e2a\u4ee4\u4eba\u4e0d\u5b89\u7684\u4f8b\u5916\uff0c\u800c\u662f\u4e00\u79cd\u5e38\u6001\u3002<code>AB \u2260 BA<\/code> \u8fd9\u4e2a\u7b80\u5355\u7684\u4e0d\u7b49\u5f0f\uff0c\u662f\u5bf9\u6211\u4eec\u5173\u4e8e\u201c\u4e58\u6cd5\u201d\u8fd9\u4e2a\u6982\u5ff5\u6700\u539f\u59cb\u76f4\u89c9\u7684\u608d\u7136\u6311\u6218\u3002\u4e00\u4e2a\u81ea\u7136\u800c\u7136\u7684\u95ee\u9898\u4fbf\u4f1a\u6d8c\u4e0a\u5fc3\u5934\uff1a<strong>\u4e3a\u4ec0\u4e48\u6211\u4eec\u8fd8\u8981\u79f0\u5176\u4e3a\u201c\u4e58\u6cd5\u201d\uff1f\u8fd9\u79cd\u770b\u4f3c\u602a\u5f02\u3001\u4e0d\u7b26\u5408\u76f4\u89c9\u7684\u8fd0\u7b97\u89c4\u5219\uff0c\u7a76\u7adf\u4ece\u4f55\u800c\u6765\uff0c\u5b83\u53c8\u63ed\u793a\u4e86\u600e\u6837\u4e00\u4e2a\u66f4\u6df1\u5c42\u6b21\u7684\u6570\u5b66\u6216\u7269\u7406\u5b9e\u5728\uff1f<\/strong><\/p>\n\n\n\n<p>\u672c\u6587\u7684\u4f7f\u547d\uff0c\u6b63\u662f\u8981\u7cfb\u7edf\u800c\u5f7b\u5e95\u5730\u56de\u7b54\u8fd9\u4e9b\u95ee\u9898\u3002\u6211\u4eec\u5c06\u4e0d\u4ec5\u4ec5\u5b66\u4e60\u5982\u4f55\u8ba1\u7b97\u77e9\u9635\u4e58\u6cd5\uff0c\u66f4\u5c06\u6df1\u5165\u63a2\u7d22\u5b83\u4e3a\u4f55\u5982\u6b64\u5b9a\u4e49\u3002\u6211\u4eec\u5c06\u4e00\u540c\u8e0f\u4e0a\u4e00\u6761\u4ece\u5177\u4f53\u5230\u62bd\u8c61\u3001\u4ece\u4ee3\u6570\u5230\u51e0\u4f55\u3001\u4ece\u7406\u8bba\u5230\u5b9e\u8df5\u7684\u65c5\u9014\u3002\u6211\u4eec\u5c06\u56de\u6eaf\u523019\u4e16\u7eaa\uff0c\u89c1\u8bc1\u5b83\u5728\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u4e2d\u7684\u8bde\u751f\uff1b\u6211\u4eec\u5c06\u6df1\u5165\u5176\u51e0\u4f55\u7075\u9b42\uff0c\u7406\u89e3\u77e9\u9635\u4f5c\u4e3a\u7a7a\u95f4\u53d8\u6362\u7684\u672c\u8d28\uff1b\u6211\u4eec\u8fd8\u5c06\u89e3\u5256\u5176\u5728\u73b0\u4ee3\u8ba1\u7b97\u673a\u4e2d\u7684\u5b9e\u73b0\uff0c\u5e76\u4e00\u7aa5\u5b83\u5982\u4f55\u6210\u4e3a\u4ece\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u5230\u4eba\u5de5\u667a\u80fd\u7b49\u5c16\u7aef\u79d1\u6280\u7684\u7edd\u5bf9\u57fa\u77f3\u3002<\/p>\n\n\n\n<p>\u8bfb\u5b8c\u672c\u6587\uff0c\u4f60\u5c06\u83b7\u5f97\u7684\u4e0d\u4ec5\u4ec5\u662f\u4e00\u4e2a\u8ba1\u7b97\u5de5\u5177\u3002\u4f60\u5c06\u83b7\u5f97\u4e00\u79cd\u5168\u65b0\u7684\u201c\u89c6\u89c9\u201d\uff0c\u4e00\u79cd\u770b\u5f85\u548c\u7406\u89e3\u7ebf\u6027\u7cfb\u7edf\u3001\u6570\u636e\u5173\u7cfb\u548c\u7a7a\u95f4\u53d8\u6362\u7684\u6570\u5b66\u76f4\u89c9\u3002\u90a3\u4e2a\u66fe\u8ba9\u5609\u8c6a\u56f0\u60d1\u7684\u4e0b\u5348\uff0c\u5c06\u6210\u4e3a\u4f60\u8ba4\u77e5\u5347\u7ea7\u7684\u8f89\u714c\u8d77\u70b9\u3002\u4f60\u5c06\u6df1\u523b\u7406\u89e3\uff0c\u4e58\u6cd5\u4e4b\u6240\u4ee5\u4e0d\u80fd\u4ea4\u6362\uff0c\u6070\u6070\u56e0\u4e3a\u5b83\u6240\u63cf\u8ff0\u7684\u4e16\u754c\u2014\u2014\u4e00\u4e2a\u7531\u6709\u5e8f\u64cd\u4f5c\u548c\u53d8\u6362\u6784\u6210\u7684\u4e16\u754c\u2014\u2014\u5176\u672c\u8eab\u5c31\u662f\u975e\u4ea4\u6362\u7684\u3002\u8fd9\u5e76\u975e\u6570\u5b66\u7684\u7f3a\u9677\uff0c\u800c\u662f\u5176\u8868\u8fbe\u80fd\u529b\u7684\u4f1f\u5927\u80dc\u5229\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>1. \u5386\u53f2\u7684\u56de\u54cd\uff1a\u4f5c\u4e3a\u7ebf\u6027\u65b9\u7a0b\u7ec4\u4f18\u96c5\u89e3\u7684\u77e9\u9635<\/strong><\/h2>\n\n\n\n<p>\u5728\u6df1\u5165\u63a2\u8ba8\u90a3\u4e9b\u4ee4\u4eba\u773c\u82b1\u7f2d\u4e71\u7684\u51e0\u4f55\u53d8\u6362\u4e4b\u524d\uff0c\u6211\u4eec\u5fc5\u987b\u5148\u56de\u5230\u77e9\u9635\u4e58\u6cd5\u6700\u539f\u59cb\u7684\u8bde\u751f\u5730\u2014\u2014\u7ebf\u6027\u4ee3\u6570\u7684\u6838\u5fc3\u95ee\u9898\uff1a\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u8fd9\u4e0d\u4ec5\u662f\u5386\u53f2\u7684\u771f\u5b9e\u8def\u5f84\uff0c\u4e5f\u662f\u7406\u89e3\u5176\u8fd0\u7b97\u89c4\u5219\u4e3a\u4f55\u5982\u6b64\u201c\u602a\u5f02\u201d\u7684\u6700\u575a\u5b9e\u7684\u57fa\u7840\u3002<\/p>\n\n\n\n<p>\u60f3\u8c61\u4e00\u4e0b19\u4e16\u7eaa\u7684\u6570\u5b66\u5bb6\uff0c\u4f8b\u5982\u9ad8\u65af\u6216\u67ef\u897f\uff0c\u4ed6\u4eec\u9762\u5bf9\u7684\u662f\u4e00\u7c7b\u53cd\u590d\u51fa\u73b0\u4e14\u6781\u4e3a\u68d8\u624b\u7684\u95ee\u9898\u3002\u6bd4\u5982\u4e0b\u9762\u8fd9\u4e2a\u7b80\u5355\u7684\u4e8c\u5143\u4e00\u6b21\u65b9\u7a0b\u7ec4\uff1a<\/p>\n\n\n\n<p>$$ \\begin{cases} 2x + 3y = 8 \\\\ 5x &#8211; 1y = 3 \\end{cases} $$<\/p>\n\n\n\n<p>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4ee3\u5165\u6d88\u5143\u6cd5\u7b49\u624b\u6bb5\u6c42\u89e3\u3002\u4f46\u5f53\u53d8\u91cf\u548c\u65b9\u7a0b\u7684\u6570\u91cf\u589e\u52a0\u5230\u4e09\u4e2a\u3001\u56db\u4e2a\uff0c\u4e43\u81f3\u6210\u767e\u4e0a\u5343\u4e2a\u65f6\uff08\u4f8b\u5982\u5728\u5929\u4f53\u529b\u5b66\u8f68\u9053\u8ba1\u7b97\u6216\u5927\u5730\u6d4b\u91cf\u4e2d\uff09\uff0c\u8fd9\u79cd\u624b\u52a8\u7684\u3001\u8fc7\u7a0b\u5316\u7684\u8ba1\u7b97\u65b9\u5f0f\u4f1a\u53d8\u5f97\u5f02\u5e38\u7e41\u7410\u4e14\u6781\u6613\u51fa\u9519\u3002\u6570\u5b66\u5bb6\u4eec\u8feb\u5207\u9700\u8981\u4e00\u79cd\u66f4\u7b80\u6d01\u3001\u66f4\u7ed3\u6784\u5316\u7684\u8868\u793a\u6cd5\u548c\u4e00\u5957\u6807\u51c6\u5316\u7684\u64cd\u4f5c\u89c4\u5219\u3002<\/p>\n\n\n\n<p>\u82f1\u56fd\u6570\u5b66\u5bb6\u963f\u745f\u00b7\u51ef\u83b1\uff08Arthur Cayley\uff09\u572819\u4e16\u7eaa\u4e2d\u53f6\u7684\u5f00\u521b\u6027\u5de5\u4f5c\uff0c\u6b63\u662f\u4e3a\u4e86\u5e94\u5bf9\u8fd9\u4e00\u6311\u6218\u3002\u4ed6\u5f15\u5165\u4e86\u201c\u77e9\u9635\u201d\u548c\u201c\u5411\u91cf\u201d\u7684\u6982\u5ff5\uff0c\u5c06\u4e0a\u8ff0\u65b9\u7a0b\u7ec4\u201c\u6253\u5305\u201d\u6210\u4e00\u79cd\u6781\u4e3a\u4f18\u96c5\u7684\u5f62\u5f0f\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/img.hcha.site\/images\/original\/portrait\/20250916_154901_3763.jpg'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  decoding=\"async\" data-original=\"https:\/\/img.hcha.site\/images\/original\/portrait\/20250916_154901_3763.jpg\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" style=\"width:250px;height:auto\"\/><\/div><figcaption class=\"wp-element-caption\"><strong>Arthur Cayley<\/strong><br>1821 \u00b7 08 \u00b7 16  &#8211;  1895 \u00b7 01 \u00b7 26<\/figcaption><\/figure>\n<\/div>\n\n\n<p>\u9996\u5148\uff0c\u6211\u4eec\u5c06\u6240\u6709\u7cfb\u6570\u62bd\u79bb\u51fa\u6765\uff0c\u5f62\u6210\u4e00\u4e2a\u6211\u4eec\u79f0\u4e4b\u4e3a\u7cfb\u6570\u77e9\u9635 <code>A<\/code> \u7684\u77e9\u5f62\u6570\u9635\uff1a<\/p>\n\n\n\n<p>$$ A = \\begin{bmatrix} 2 &amp; 3 \\\\ 5 &amp; -1 \\end{bmatrix} $$<\/p>\n\n\n\n<p>\u7136\u540e\uff0c\u6211\u4eec\u5c06\u672a\u77e5\u6570 <code>x<\/code> \u548c <code>y<\/code> \u7ec4\u7ec7\u6210\u4e00\u4e2a\u5217\u5411\u91cf <code>x<\/code>\uff08\u5411\u91cf\u53ef\u4ee5\u770b\u4f5c\u662f\u53ea\u6709\u4e00\u5217\u7684\u7279\u6b8a\u77e9\u9635\uff09\uff1a<\/p>\n\n\n\n<p>$$ \\mathbf{x} = \\begin{bmatrix} x \\\\ y \\end{bmatrix} $$<\/p>\n\n\n\n<p>\u6700\u540e\uff0c\u6211\u4eec\u5c06\u65b9\u7a0b\u53f3\u4fa7\u7684\u5e38\u6570\u4e5f\u7ec4\u7ec7\u6210\u4e00\u4e2a\u5217\u5411\u91cf <code>b<\/code>\uff1a<\/p>\n\n\n\n<p>$$ \\mathbf{b} = \\begin{bmatrix} 8 \\\\ 3 \\end{bmatrix} $$<\/p>\n\n\n\n<p>\u73b0\u5728\uff0c\u6574\u4e2a\u65b9\u7a0b\u7ec4\u88ab\u7b80\u6d01\u5730\u8868\u8fbe\u4e3a\uff1a<\/p>\n\n\n\n<p>$$ A\\mathbf{x} = \\mathbf{b} $$<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u8868\u8fbe\u5f0f\u770b\u8d77\u6765\u5e72\u51c0\u5229\u843d\uff0c\u4f46\u5b83\u76ee\u524d\u8fd8\u53ea\u662f\u4e00\u4e2a\u7b26\u53f7\u4e0a\u7684\u7ea6\u5b9a\u3002\u5b83\u7684\u5a01\u529b\u5728\u4e8e\uff0c\u6211\u4eec<strong>\u5fc5\u987b<\/strong>\u53bb\u5b9a\u4e49\u4e00\u79cd\u77e9\u9635\u4e0e\u5411\u91cf\u4e4b\u95f4\u7684\u201c\u4e58\u6cd5\u201d\uff0c\u4f7f\u5f97 <code>Ax<\/code> \u7684\u8ba1\u7b97\u7ed3\u679c\u6070\u597d\u80fd\u8fd8\u539f\u51fa\u539f\u59cb\u65b9\u7a0b\u7ec4\u7684\u5de6\u4fa7\u3002\u8ba9\u6211\u4eec\u770b\u770b\u9700\u8981\u53d1\u751f\u4ec0\u4e48\uff1a<\/p>\n\n\n\n<p>$$ \\begin{bmatrix} 2 &amp; 3 \\\\ 5 &amp; -1 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} = \\begin{bmatrix} 2x + 3y \\\\ 5x &#8211; 1y \\end{bmatrix} $$<\/p>\n\n\n\n<p>\u4e3a\u4e86\u8ba9\u8fd9\u4e2a\u7b49\u5f0f\u6210\u7acb\uff0c\u77e9\u9635\u4e0e\u5411\u91cf\u7684\u4e58\u6cd5\u5fc5\u987b\u9075\u5faa\u4e00\u4e2a\u7279\u5b9a\u7684\u6a21\u5f0f\uff1a\u7ed3\u679c\u5411\u91cf\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20\uff08<code>2x + 3y<\/code>\uff09\uff0c\u662f\u7531\u77e9\u9635 <code>A<\/code> \u7684\u7b2c\u4e00\u884c <code>[2, 3]<\/code> \u4e0e\u5411\u91cf <code>x<\/code> \u7684\u5217 <code>[x, y]<\/code> \u7684\u5bf9\u5e94\u5143\u7d20\u76f8\u4e58\u518d\u76f8\u52a0\u5f97\u5230\u7684\u3002\u540c\u6837\uff0c\u7ed3\u679c\u5411\u91cf\u7684\u7b2c\u4e8c\u4e2a\u5143\u7d20\uff08<code>5x - 1y<\/code>\uff09\uff0c\u662f\u7531\u77e9\u9635 <code>A<\/code> \u7684\u7b2c\u4e8c\u884c <code>[5, -1]<\/code> \u4e0e\u5411\u91cf <code>x<\/code> \u7684\u5217\u5bf9\u5e94\u76f8\u4e58\u518d\u76f8\u52a0\u5f97\u5230\u7684\u3002<\/p>\n\n\n\n<p>\u8fd9\u4e2a<strong>\u201c\u884c\u4e58\u4ee5\u5217\u201d<\/strong>\u7684\u89c4\u5219\uff0c\u5e76\u975e\u51ed\u7a7a\u634f\u9020\uff0c\u800c\u662f\u4e3a\u4e86\u4fdd\u6301 <code>Ax=b<\/code> \u8fd9\u4e00\u8bb0\u6cd5\u4e0e\u539f\u59cb\u65b9\u7a0b\u7ec4\u5728\u903b\u8f91\u4e0a\u5b8c\u5168\u7b49\u4ef7\u6240<strong>\u5fc5\u7136<\/strong>\u5bfc\u51fa\u7684\u7ed3\u679c\u3002\u5b83\u662f\u9700\u6c42\u9a71\u52a8\u7684\u4ea7\u7269\uff0c\u800c\u975e\u968f\u610f\u7684\u53d1\u660e\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>\u6df1\u5ea6\u6c42\u7d22\uff1a\u4ece\u53d8\u91cf\u66ff\u6362\u5230\u77e9\u9635\u4e58\u6cd5<\/strong><\/h4>\n\n\n\n<p>\u73b0\u5728\uff0c\u8ba9\u6211\u4eec\u628a\u60c5\u51b5\u53d8\u5f97\u66f4\u590d\u6742\u4e00\u4e9b\uff0c\u8fd9\u4e5f\u662f\u63ed\u793a\u77e9\u9635\u4e0e\u77e9\u9635\u76f8\u4e58\u5b9a\u4e49\u7684\u5173\u952e\u4e00\u6b65\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u7ec4\u53d8\u91cf <code>x, y<\/code>\uff0c\u5b83\u4eec\u7531\u53e6\u4e00\u7ec4\u53d8\u91cf <code>u, v<\/code> \u7ebf\u6027\u8868\u793a\uff1a<\/p>\n\n\n\n<p>$$ \\begin{cases} x = 7u + 1v \\\\ y = 4u + 9v \\end{cases} $$<\/p>\n\n\n\n<p>\u540c\u65f6\uff0c\u6211\u4eec\u8fd8\u6709\u7b2c\u4e09\u7ec4\u53d8\u91cf <code>z1, z2<\/code>\uff0c\u5b83\u4eec\u7531 <code>x, y<\/code> \u7ebf\u6027\u8868\u793a\uff1a<\/p>\n\n\n\n<p>$$ \\begin{cases} z_1 = 2x + 3y \\\\ z_2 = 5x &#8211; 1y \\end{cases} $$<\/p>\n\n\n\n<p>\u4e00\u4e2a\u81ea\u7136\u7684\u95ee\u9898\u662f\uff1a<code>z1, z2<\/code> \u5982\u4f55\u7531 <code>u, v<\/code> \u76f4\u63a5\u8868\u793a\uff1f\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u7eaf\u7cb9\u7684\u4ee3\u6570\u66ff\u6362\uff1a<\/p>\n\n\n\n<p>\u5c06 <code>x<\/code> \u548c <code>y<\/code> \u7684\u8868\u8fbe\u5f0f\u4ee3\u5165\u5230 <code>z1<\/code> \u548c <code>z2<\/code> \u7684\u8868\u8fbe\u5f0f\u4e2d\uff1a<\/p>\n\n\n\n<p><code>z1 = 2(7u + 1v) + 3(4u + 9v) = (2*7 + 3*4)u + (2*1 + 3*9)v = 26u + 29v<\/code><br><code>z2 = 5(7u + 1v) - 1(4u + 9v) = (5*7 - 1*4)u + (5*1 - 1*9)v = 31u - 4v<\/code><\/p>\n\n\n\n<p>\u73b0\u5728\uff0c\u8ba9\u6211\u4eec\u7528\u77e9\u9635\u7684\u8bed\u8a00\u6765\u91cd\u8ff0\u6574\u4e2a\u8fc7\u7a0b\u3002<\/p>\n\n\n\n<p><code>z<\/code> \u4e0e <code>x,y<\/code> \u7684\u5173\u7cfb\u53ef\u4ee5\u5199\u6210 <code>z = Ax<\/code>\uff1a<br>$$ \\begin{bmatrix} z_1 \\\\ z_2 \\end{bmatrix} = \\begin{bmatrix} 2 &amp; 3 \\\\ 5 &amp; -1 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} $$<br>\u5176\u4e2d <code>A<\/code> \u662f\u6211\u4eec\u719f\u6089\u7684\u90a3\u4e2a\u77e9\u9635\u3002<\/p>\n\n\n\n<p><code>x,y<\/code> \u4e0e <code>u,v<\/code> \u7684\u5173\u7cfb\u53ef\u4ee5\u5199\u6210 <code>x = Bu<\/code>\uff1a<br>$$ \\begin{bmatrix} x \\\\ y \\end{bmatrix} = \\begin{bmatrix} 7 &amp; 1 \\\\ 4 &amp; 9 \\end{bmatrix} \\begin{bmatrix} u \\\\ v \\end{bmatrix} $$<br>\u5176\u4e2d <code>B<\/code> \u662f\u65b0\u7684\u7cfb\u6570\u77e9\u9635\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u60f3\u8981\u627e\u5230\u4e00\u4e2a\u77e9\u9635 <code>C<\/code>\uff0c\u4f7f\u5f97 <code>z = Cu<\/code>\uff0c\u5373\u76f4\u63a5\u5efa\u7acb <code>z<\/code> \u548c <code>u<\/code> \u7684\u5173\u7cfb\u3002\u4ece\u4ee3\u6570\u4e0a\u770b\uff0c\u6211\u4eec\u6709 <code>z = A(Bu) = (AB)u<\/code>\uff0c\u6240\u4ee5 <code>C<\/code> \u5e94\u8be5\u5c31\u662f <code>A<\/code> \u548c <code>B<\/code> \u7684\u201c\u4e58\u79ef\u201d\u3002<\/p>\n\n\n\n<p>\u8ba9\u6211\u4eec\u770b\u770b <code>C<\/code> \u5e94\u8be5\u662f\u4ec0\u4e48\u6837\u5b50\u3002\u6839\u636e\u6211\u4eec\u521a\u521a\u8d39\u529b\u8ba1\u7b97\u51fa\u7684\u7ed3\u679c\uff1a<\/p>\n\n\n\n<p>$$ \\begin{cases} z_1 = 26u + 29v \\\\ z_2 = 31u &#8211; 4v \\end{cases} \\Rightarrow C = \\begin{bmatrix} 26 &amp; 29 \\\\ 31 &amp; -4 \\end{bmatrix} $$<\/p>\n\n\n\n<p>\u73b0\u5728\uff0c\u6211\u4eec\u9762\u4e34\u7740\u5b9a\u4e49\u77e9\u9635\u4e58\u6cd5\u7684\u201c\u5ba1\u5224\u65f6\u523b\u201d\u3002\u6211\u4eec\u5fc5\u987b\u5b9a\u4e49\u4e00\u79cd <code>A<\/code> \u548c <code>B<\/code> \u4e4b\u95f4\u7684\u4e58\u6cd5\u8fd0\u7b97\uff0c\u4f7f\u5f97 <code>AB<\/code> \u7684\u7ed3\u679c\u6070\u597d\u662f <code>C<\/code>\u3002<\/p>\n\n\n\n<p>$$ A \\times B = \\begin{bmatrix} 2 &amp; 3 \\\\ 5 &amp; -1 \\end{bmatrix} \\times \\begin{bmatrix} 7 &amp; 1 \\\\ 4 &amp; 9 \\end{bmatrix} = \\begin{bmatrix} ? &amp; ? \\\\ ? &amp; ? \\end{bmatrix} $$<\/p>\n\n\n\n<p>\u8ba9\u6211\u4eec\u805a\u7126\u4e8e\u7ed3\u679c <code>C<\/code> \u7684\u5de6\u4e0a\u89d2\u5143\u7d20 <code>26<\/code>\u3002\u56de\u987e\u6211\u4eec\u7684\u4ee3\u6570\u63a8\u5bfc\uff0c<code>26<\/code> \u6765\u81ea\u4e8e <code>(2*7 + 3*4)<\/code>\u3002\u8bf7\u6ce8\u610f\u770b\u8fd9\u4e9b\u6570\u5b57\u7684\u6765\u6e90\uff1a<code>[2, 3]<\/code> \u6b63\u662f\u77e9\u9635 <code>A<\/code> \u7684\u7b2c\u4e00\u884c\uff0c\u800c <code>[7, 4]<\/code> \u6b63\u662f\u77e9\u9635 <code>B<\/code> \u7684\u7b2c\u4e00\u5217\uff01<\/p>\n\n\n\n<p>\u518d\u770b <code>C<\/code> \u7684\u53f3\u4e0a\u89d2\u5143\u7d20 <code>29<\/code>\u3002\u5b83\u6765\u81ea\u4e8e <code>(2*1 + 3*9)<\/code>\u3002<code>[2, 3]<\/code> \u662f <code>A<\/code> \u7684\u7b2c\u4e00\u884c\uff0c\u800c <code>[1, 9]<\/code> \u662f <code>B<\/code> \u7684\u7b2c\u4e8c\u5217\u3002<\/p>\n\n\n\n<p>\u89c4\u5f8b\u5df2\u7ecf\u662d\u7136\u82e5\u63ed\uff01\u7ed3\u679c\u77e9\u9635 <code>C<\/code> \u4e2d\u7b2c <code>i<\/code> \u884c\u3001\u7b2c <code>j<\/code> \u5217\u7684\u5143\u7d20 <code>C[i, j]<\/code>\uff0c\u6b63\u662f\u7531\u77e9\u9635 <code>A<\/code> \u7684\u7b2c <code>i<\/code> \u884c\u4e0e\u77e9\u9635 <code>B<\/code> \u7684\u7b2c <code>j<\/code> \u5217\uff0c\u8fdb\u884c\u5bf9\u5e94\u5143\u7d20\u76f8\u4e58\u518d\u6c42\u548c\uff08\u8fd9\u4e2a\u8fc7\u7a0b\u4e5f\u88ab\u79f0\u4e3a\u70b9\u79ef\u6216\u5185\u79ef\uff09\u5f97\u5230\u7684\u3002<\/p>\n\n\n\n<p>\u8fd9\uff0c\u5c31\u662f\u77e9\u9635\u4e58\u6cd5\u90a3\u6761\u8457\u540d\u4f46\u5e38\u5e38\u663e\u5f97\u6709\u4e9b\u795e\u79d8\u7684\u201c\u884c\u4e58\u4ee5\u5217\u201d\u89c4\u5219\u7684\u4ee3\u6570\u8d77\u6e90\u3002\u5b83\u4e0d\u662f\u4e00\u4e2a\u6b66\u65ad\u7684\u89c4\u5b9a\uff0c\u800c\u662f\u4e3a\u4e86\u4f7f<strong>\u201c\u53d8\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5176\u7cfb\u6570\u77e9\u9635\u662f\u539f\u7cfb\u6570\u77e9\u9635\u7684\u4e58\u79ef\u201d<\/strong>\u8fd9\u4e00\u7f8e\u5999\u7684\u6027\u8d28\u5f97\u4ee5\u6210\u7acb\uff0c\u6240\u5fc5\u987b\u91c7\u53d6\u7684\u552f\u4e00\u5408\u7406\u7684\u5b9a\u4e49\u3002\u5b83\u5b8c\u7f8e\u5730\u5c06\u7e41\u7410\u7684\u4ee3\u6570\u66ff\u6362\u64cd\u4f5c\uff0c\u5c01\u88c5\u8fdb\u4e86\u4e00\u5957\u5e72\u51c0\u3001\u7edf\u4e00\u7684\u77e9\u9635\u8fd0\u7b97\u6846\u67b6\u4e2d\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>2. \u51e0\u4f55\u7684\u76f4\u89c9\uff1a\u4f5c\u4e3a\u7a7a\u95f4\u53d8\u6362\u6307\u4ee4\u7684\u77e9\u9635<\/strong><\/h2>\n\n\n\n<p>\u5982\u679c\u8bf4\u4e0a\u4e00\u8282\u6211\u4eec\u4ece\u4ee3\u6570\u5c42\u9762\u5efa\u7acb\u4e86\u77e9\u9635\u4e58\u6cd5\u89c4\u5219\u7684\u5408\u6cd5\u6027\uff0c\u90a3\u4e48\u8fd9\u4e00\u8282\uff0c\u6211\u4eec\u5c06\u4e3a\u5b83\u6ce8\u5165\u7075\u9b42\u3002\u6211\u4eec\u5c06\u63ed\u793a\uff0c\u77e9\u9635\u8fdc\u4e0d\u6b62\u662f\u7cfb\u6570\u7684\u5bb9\u5668\uff0c\u5b83\u672c\u8d28\u4e0a\u662f\u7ebf\u6027\u53d8\u6362\u7684\u51e0\u4f55\u5316\u8eab\u3002\u800c\u77e9\u9635\u4e58\u6cd5\uff0c\u6b63\u662f\u53d8\u6362\u7684\u590d\u5408\u3002<\/p>\n\n\n\n<p>\u60f3\u8c61\u4e00\u4e2a\u4e8c\u7ef4\u5e73\u9762\uff0c\u4e0a\u9762\u5e03\u6ee1\u4e86\u65e0\u6570\u7684\u5411\u91cf\u3002\u4e00\u4e2a\u53d8\u6362\u5c31\u662f\u4e00\u4e2a\u51fd\u6570\uff0c\u5b83\u63a5\u6536\u4e00\u4e2a\u5411\u91cf\u4f5c\u4e3a\u8f93\u5165\uff0c\u7136\u540e\u8f93\u51fa\u4e00\u4e2a\u53d8\u6362\u540e\u7684\u65b0\u5411\u91cf\u3002\u6240\u8c13\u7ebf\u6027\u53d8\u6362\uff0c\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u3001\u8868\u73b0\u826f\u597d\u7684\u53d8\u6362\uff0c\u5b83\u4fdd\u6301\u4e86\u7f51\u683c\u7ebf\u7684\u5e73\u884c\u548c\u7b49\u8ddd\u3002\u7528\u66f4\u4e25\u683c\u7684\u6570\u5b66\u8bed\u8a00\u6765\u8bf4\uff0c\u4e00\u4e2a\u53d8\u6362 <code>T<\/code>\u662f\u7ebf\u6027\u7684\uff0c\u5982\u679c\u5b83\u6ee1\u8db3\u4ee5\u4e0b\u4e24\u4e2a\u6761\u4ef6\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><code>T(v + w) = T(v) + T(w)<\/code> (\u53ef\u52a0\u6027\uff1a\u4e24\u4e2a\u5411\u91cf\u76f8\u52a0\u540e\u7684\u53d8\u6362\uff0c\u7b49\u4e8e\u5b83\u4eec\u5404\u81ea\u53d8\u6362\u540e\u518d\u76f8\u52a0)<\/li>\n\n\n\n<li><code>T(c*v) = c*T(v)<\/code> (\u9f50\u6b21\u6027\uff1a\u4e00\u4e2a\u5411\u91cf\u4f38\u7f29c\u500d\u540e\u7684\u53d8\u6362\uff0c\u7b49\u4e8e\u5b83\u53d8\u6362\u540e\u518d\u4f38\u7f29c\u500d)<\/li>\n<\/ol>\n\n\n\n<p>\u73b0\u5728\uff0c\u8ba9\u6211\u4eec\u601d\u8003\u4e00\u4e2a\u5173\u952e\u95ee\u9898\uff1a\u5982\u4f55\u7528\u6570\u5b57\u6765\u63cf\u8ff0\u4e00\u4e2a\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff1f\u7b54\u6848\u51fa\u5947\u7684\u7b80\u5355\uff1a\u6211\u4eec\u53ea\u9700\u8981\u8ffd\u8e2a\u8fd9\u4e2a\u53d8\u6362\u5bf9<strong>\u57fa\u5411\u91cf<\/strong>\u505a\u4e86\u4ec0\u4e48\u3002<\/p>\n\n\n\n<p>\u5728\u6807\u51c6\u7684\u4e8c\u7ef4\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e2d\uff0c\u6211\u4eec\u7684\u57fa\u5411\u91cf\u662f\uff1a<br><code>\u00ee = [1, 0]<\/code> (\u6cbfx\u8f74\u7684\u5355\u4f4d\u5411\u91cf)<br><code>\u0135 = [0, 1]<\/code> (\u6cbfy\u8f74\u7684\u5355\u4f4d\u5411\u91cf)<\/p>\n\n\n\n<p>\u4efb\u4f55\u4e00\u4e2a\u5411\u91cf <code>v = [x, y]<\/code> \u90fd\u53ef\u4ee5\u8868\u793a\u4e3a\u8fd9\u4e24\u4e2a\u57fa\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff1a<code>v = x*\u00ee + y*\u0135<\/code>\u3002<\/p>\n\n\n\n<p>\u7531\u4e8e\u7ebf\u6027\u53d8\u6362\u7684\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u53d8\u6362\u540e\u7684\u5411\u91cf <code>T(v)<\/code> \u662f\u4ec0\u4e48\uff1a<br><code>T(v) = T(x*\u00ee + y*\u0135) = T(x*\u00ee) + T(y*\u0135) = x*T(\u00ee) + y*T(\u0135)<\/code><\/p>\n\n\n\n<p>\u8fd9\u4e2a\u516c\u5f0f\u662f\u7406\u89e3\u77e9\u9635\u51e0\u4f55\u610f\u4e49\u7684\u6839\u57fa\uff01\u5b83\u544a\u8bc9\u6211\u4eec\uff1a\u53ea\u8981\u6211\u4eec\u77e5\u9053\u53d8\u6362\u540e\u7684\u57fa\u5411\u91cf <code>T(\u00ee)<\/code> \u548c <code>T(\u0135)<\/code> \u5728\u54ea\u91cc\uff0c\u6211\u4eec\u5c31\u80fd\u7acb\u523b\u786e\u5b9a\u4efb\u4f55\u4e00\u4e2a\u5411\u91cf <code>v=[x,y]<\/code> \u7ecf\u8fc7\u53d8\u6362\u540e\u7684\u4f4d\u7f6e\u3002<\/p>\n\n\n\n<p>\u5047\u8bbe\u67d0\u4e2a\u7ebf\u6027\u53d8\u6362 <code>L<\/code> \u628a <code>\u00ee<\/code> \u53d8\u6362\u5230\u4e86 <code>[a, c]<\/code>\uff0c\u628a <code>\u0135<\/code> \u53d8\u6362\u5230\u4e86 <code>[b, d]<\/code>\u3002<br>\u5373\uff1a<code>L(\u00ee) = [a, c]<\/code>\uff0c<code>L(\u0135) = [b, d]<\/code>\u3002<br>\u90a3\u4e48\uff0c\u5411\u91cf <code>v=[x,y]<\/code> \u53d8\u6362\u540e\u7684\u7ed3\u679c <code>L(v)<\/code> \u5c31\u662f\uff1a<br><code>L(v) = x*[a, c] + y*[b, d] = [xa + yb, xc + yd]<\/code><\/p>\n\n\n\n<p>\u73b0\u5728\uff0c\u8ba9\u6211\u4eec\u6784\u9020\u4e00\u4e2a\u77e9\u9635 <code>M<\/code>\uff0c\u5b83\u7684\u7b2c\u4e00\u5217\u5c31\u662f\u53d8\u6362\u540e\u7684 <code>\u00ee<\/code>\uff0c\u7b2c\u4e8c\u5217\u5c31\u662f\u53d8\u6362\u540e\u7684 <code>\u0135<\/code>\uff1a<br>$$ M = \\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix} $$<br>\u7136\u540e\uff0c\u6211\u4eec\u6765\u8ba1\u7b97\u77e9\u9635 <code>M<\/code> \u4e0e\u5411\u91cf <code>v<\/code> \u7684\u4e58\u79ef <code>Mv<\/code> (\u4f7f\u7528\u6211\u4eec\u5728\u7b2c\u4e00\u8282\u4e2d\u63a8\u5bfc\u51fa\u7684\u884c-\u5217\u89c4\u5219)\uff1a<br>$$ Mv = \\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} = \\begin{bmatrix} ax + by \\\\ cx + dy \\end{bmatrix} $$<br>\u770b\u5230\u4e86\u5417\uff1f<code>Mv<\/code> \u7684\u8ba1\u7b97\u7ed3\u679c\u4e0e\u6211\u4eec\u4ece\u51e0\u4f55\u4e0a\u63a8\u5bfc\u51fa\u7684 <code>L(v)<\/code> \u5b8c\u5168\u4e00\u6837\uff01<\/p>\n\n\n\n<p>\u8fd9\u662f\u4e00\u4e2a\u6781\u5176\u91cd\u8981\u7684\u89c2\u5bdf\uff1a<strong>\u4e00\u4e2a2&#215;2\u77e9\u9635\uff0c\u5c31\u662f\u5bf9\u4e8c\u7ef4\u7a7a\u95f4\u8fdb\u884c\u4e00\u6b21\u7ebf\u6027\u53d8\u6362\u7684\u6307\u4ee4\u3002\u5b83\u7684\u5217\uff0c\u660e\u786e\u5730\u544a\u8bc9\u6211\u4eec\uff0c\u539f\u6765\u7684\u57fa\u5411\u91cf \u00ee \u548c \u0135 \u4f1a\u88ab\u53d8\u6362\u5230\u54ea\u91cc\u53bb\u3002\u800c\u77e9\u9635\u4e58\u4ee5\u4e00\u4e2a\u5411\u91cf\uff0c\u5c31\u662f\u5728\u51e0\u4f55\u610f\u4e49\u4e0a\u5c06\u90a3\u4e2a\u53d8\u6362\u65bd\u52a0\u4e8e\u8be5\u5411\u91cf\u3002<\/strong><\/p>\n\n\n\n<p>\u4f8b\u5982\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u65cb\u8f6c\u77e9\u9635<\/strong>\uff1a\u9006\u65f6\u9488\u65cb\u8f6c90\u5ea6\u7684\u53d8\u6362\uff0c\u4f1a\u628a <code>\u00ee=[1,0]<\/code> \u53d8\u5230 <code>[0,1]<\/code>\uff0c\u628a <code>\u0135=[0,1]<\/code> \u53d8\u5230 <code>[-1,0]<\/code>\u3002\u6240\u4ee5\uff0c\u8fd9\u4e2a\u53d8\u6362\u5bf9\u5e94\u7684\u77e9\u9635\u5c31\u662f <code>[[0, -1], [1, 0]]<\/code>\u3002<\/li>\n\n\n\n<li><strong>\u526a\u5207\u77e9\u9635<\/strong>\uff1a\u4e00\u4e2a\u6c34\u5e73\u526a\u5207\u53d8\u6362\uff0c\u4fdd\u6301 <code>\u00ee<\/code> \u4e0d\u53d8\uff0c\u4f46\u628a <code>\u0135=[0,1]<\/code> \u63a8\u5230 <code>[1,1]<\/code>\u3002\u8fd9\u4e2a\u53d8\u6362\u7684\u77e9\u9635\u5c31\u662f <code>[[1, 1], [0, 1]]<\/code>\u3002<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u51e0\u4f55\u76f4\u89c9\u4e0b\u7684\u77e9\u9635\u4e58\u6cd5\uff1a\u53d8\u6362\u7684\u63a5\u7eed\u4e0a\u6f14<\/strong><\/h3>\n\n\n\n<p>\u6709\u4e86\u8fd9\u4e2a\u5f3a\u5927\u7684\u51e0\u4f55\u76f4\u89c9\uff0c\u6211\u4eec\u7ec8\u4e8e\u53ef\u4ee5\u4ece\u4e00\u4e2a\u5168\u65b0\u7684\u3001\u66f4\u6839\u672c\u7684\u89c6\u89d2\u6765\u7406\u89e3\u77e9\u9635\u4e58\u6cd5\u7684\u771f\u8c1b\uff0c\u5e76\u4e00\u4e3e\u653b\u514b\u201c\u975e\u4ea4\u6362\u6027\u201d\u7684\u8c1c\u9898\u3002<\/p>\n\n\n\n<p><strong>\u5982\u679c\u5355\u4e2a\u77e9\u9635\u4ee3\u8868\u4e00\u6b21\u53d8\u6362\uff0c\u90a3\u4e48\u4e24\u4e2a\u77e9\u9635\u76f8\u4e58 <code>A \u00d7 B<\/code> \u4ee3\u8868\u4ec0\u4e48\uff1f<\/strong><\/p>\n\n\n\n<p>\u7b54\u6848\u662f\uff1a<strong>\u4ee3\u8868\u4e24\u4e2a\u53d8\u6362\u7684\u8fde\u7eed\u65bd\u52a0\u3002<\/strong> \u66f4\u5177\u4f53\u5730\u8bf4\uff0c<code>AB<\/code>\u6240\u4ee3\u8868\u7684\u590d\u5408\u53d8\u6362\uff0c\u662f\u201c\u5148\u65bd\u52a0\u53d8\u6362 <code>B<\/code>\uff0c\u7136\u540e\u65bd\u52a0\u53d8\u6362 <code>A<\/code>\u201d\u3002\uff08\u6ce8\u610f\u8fd9\u4e2a\u987a\u5e8f\uff0c\u8fd9\u5728\u6570\u5b66\u4e0a\u79f0\u4e3a\u51fd\u6570\u590d\u5408\uff0c<code>g(f(x))<\/code> \u5199\u4f5c <code>(g\u2218f)(x)<\/code>\uff0c<code>f<\/code> \u5148\u4f5c\u7528\uff09\u3002<\/p>\n\n\n\n<p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5411\u91cf <code>v<\/code>\uff0c\u6211\u4eec\u60f3\u5bf9\u5b83\u5148\u505a <code>B<\/code> \u53d8\u6362\uff0c\u518d\u505a <code>A<\/code> \u53d8\u6362\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u7ecf\u8fc7 <code>B<\/code> \u53d8\u6362\uff0c<code>v<\/code> \u53d8\u6210\u4e86 <code>Bv<\/code>\u3002\u6211\u4eec\u79f0\u8fd9\u4e2a\u65b0\u5411\u91cf\u4e3a <code>v'<\/code>\u3002<\/li>\n\n\n\n<li>\u63a5\u7740\uff0c\u6211\u4eec\u5bf9 <code>v'<\/code> \u518d\u8fdb\u884c <code>A<\/code> \u53d8\u6362\uff0c\u7ed3\u679c\u662f <code>A(v')<\/code>\uff0c\u4e5f\u5c31\u662f <code>A(Bv)<\/code>\u3002<\/li>\n\n\n\n<li>\u6839\u636e\u77e9\u9635\u4e58\u6cd5\u7684\u7ed3\u5408\u5f8b\uff0c<code>A(Bv)<\/code> \u7b49\u4e8e <code>(AB)v<\/code>\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u8fd9\u8bf4\u660e\uff0c\u77e9\u9635\u4e58\u79ef <code>AB<\/code> \u8fd9\u4e2a\u5355\u4e00\u7684\u77e9\u9635\uff0c\u5b83\u6240\u4ee3\u8868\u7684\u53d8\u6362\u6548\u679c\uff0c\u4e0e\u201c\u5148\u5e94\u7528 <code>B<\/code> \u53d8\u6362\uff0c\u518d\u5e94\u7528 <code>A<\/code> \u53d8\u6362\u201d\u7684\u7ec4\u5408\u6548\u679c\u662f\u5b8c\u5168\u4e00\u6837\u7684\u3002<code>AB<\/code> \u5c31\u662f\u8fd9\u4e2a\u590d\u5408\u53d8\u6362\u7684\u201c\u5feb\u6377\u65b9\u5f0f\u201d\u6216\u201c\u6d53\u7f29\u6307\u4ee4\u201d\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u975e\u4ea4\u6362\u6027\u7684\u8c41\u7136\u5f00\u6717<\/strong><\/h3>\n\n\n\n<p>\u73b0\u5728\uff0c\u90a3\u4e2a\u56f0\u6270\u5609\u8c6a\u7684\u95ee\u9898\u2014\u2014\u4e3a\u4ec0\u4e48 <code>AB \u2260 BA<\/code> \u2014\u2014\u7ec8\u4e8e\u6709\u4e86\u6e05\u6670\u3001\u76f4\u89c2\u7684\u51e0\u4f55\u89e3\u91ca\u3002<\/p>\n\n\n\n<p><code>ABv<\/code> \u610f\u5473\u7740\u5bf9v\u201c\u5148B\uff0c\u518dA\u201d\u3002<br><code>BAv<\/code> \u610f\u5473\u7740\u5bf9v\u201c\u5148A\uff0c\u518dB\u201d\u3002<\/p>\n\n\n\n<p><strong>\u5728\u7a7a\u95f4\u53d8\u6362\u7684\u4e16\u754c\u91cc\uff0c\u64cd\u4f5c\u7684\u987a\u5e8f\u81f3\u5173\u91cd\u8981\u3002<\/strong><\/p>\n\n\n\n<p>\u8ba9\u6211\u4eec\u6765\u770b\u4e00\u4e2a\u7ecf\u5178\u7684\u4f8b\u5b50\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u4ee4 <code>A<\/code> \u4e3a\u9006\u65f6\u9488\u65cb\u8f6c90\u5ea6\u7684\u53d8\u6362\uff1a<code>A = [[0, -1], [1, 0]]<\/code><\/li>\n\n\n\n<li>\u4ee4 <code>B<\/code> \u4e3a\u6c34\u5e73\u526a\u5207\u53d8\u6362\uff1a<code>B = [[1, 1], [0, 1]]<\/code><\/li>\n<\/ul>\n\n\n\n<p>\u8ba9\u6211\u4eec\u770b\u770b <code>AB<\/code> \u548c <code>BA<\/code> \u5206\u522b\u662f\u4ec0\u4e48\uff1a<\/p>\n\n\n\n<p><strong>1. \u8ba1\u7b97 <code>AB<\/code> (\u5148\u526a\u5207\uff0c\u518d\u65cb\u8f6c)<\/strong><br>$$ AB = \\begin{bmatrix} 0 &amp; -1 \\\\ 1 &amp; 0 \\end{bmatrix} \\begin{bmatrix} 1 &amp; 1 \\\\ 0 &amp; 1 \\end{bmatrix} = \\begin{bmatrix} (0 \\cdot 1 + (-1) \\cdot 0) &amp; (0 \\cdot 1 + (-1) \\cdot 1) \\\\ (1 \\cdot 1 + 0 \\cdot 0) &amp; (1 \\cdot 1 + 0 \\cdot 1) \\end{bmatrix} = \\begin{bmatrix} 0 &amp; -1 \\\\ 1 &amp; 1 \\end{bmatrix} $$<br>\u51e0\u4f55\u4e0a\uff0c\u5148\u526a\u5207\u518d\u65cb\u8f6c\uff0c\u4f1a\u628a <code>\u00ee<\/code> \u53d8\u5230 <code>[0,1]<\/code>\uff0c\u628a <code>\u0135<\/code> \u53d8\u5230 <code>[-1,1]<\/code>\u3002\u8fd9\u6b63\u662f <code>AB<\/code> \u77e9\u9635\u7684\u5217\uff01<\/p>\n\n\n\n<p><strong>2. \u8ba1\u7b97 <code>BA<\/code> (\u5148\u65cb\u8f6c\uff0c\u518d\u526a\u5207)<\/strong><br>$$ BA = \\begin{bmatrix} 1 &amp; 1 \\\\ 0 &amp; 1 \\end{bmatrix} \\begin{bmatrix} 0 &amp; -1 \\\\ 1 &amp; 0 \\end{bmatrix} = \\begin{bmatrix} (1 \\cdot 0 + 1 \\cdot 1) &amp; (1 \\cdot (-1) + 1 \\cdot 0) \\\\ (0 \\cdot 0 + 1 \\cdot 1) &amp; (0 \\cdot (-1) + 1 \\cdot 0) \\end{bmatrix} = \\begin{bmatrix} 1 &amp; -1 \\\\ 1 &amp; 0 \\end{bmatrix} $$<br>\u51e0\u4f55\u4e0a\uff0c\u5148\u65cb\u8f6c\u518d\u526a\u5207\uff0c\u4f1a\u628a <code>\u00ee<\/code> \u53d8\u5230 <code>[1,1]<\/code>\uff0c\u628a <code>\u0135<\/code> \u53d8\u5230 <code>[-1,0]<\/code>\u3002\u8fd9\u6b63\u662f <code>BA<\/code> \u77e9\u9635\u7684\u5217\uff01<\/p>\n\n\n\n<p>\u663e\u7136\uff0c<code>AB \u2260 BA<\/code>\u3002\u56e0\u4e3a\u201c\u5148\u7a7f\u889c\u5b50\u518d\u7a7f\u978b\u201d\u548c\u201c\u5148\u7a7f\u978b\u518d\u7a7f\u889c\u5b50\u201d\u662f\u622a\u7136\u4e0d\u540c\u7684\u7ed3\u679c\u3002\u77e9\u9635\u4e58\u6cd5\u7684\u4e0d\u4ea4\u6362\u6027\uff0c\u4e0d\u662f\u4e00\u4e2a\u7f3a\u9677\uff0c\u800c\u662f\u5b83\u5fe0\u5b9e\u5730\u53cd\u6620\u4e86\u6211\u4eec\u751f\u6d3b\u7684\u8fd9\u4e2a\u4e16\u754c\u4e2d\uff0c\u64cd\u4f5c\u987a\u5e8f\u8fd9\u4e00\u57fa\u672c\u73b0\u5b9e\u3002\u4ece\u51b2\u6ce1\u5496\u5561\u5230\u7f16\u7a0b\uff0c\u4ece\u5316\u5b66\u53cd\u5e94\u5230\u91cf\u5b50\u529b\u5b66\u7684\u6d4b\u91cf\uff0c\u987a\u5e8f\u51b3\u5b9a\u4e00\u5207\u3002\u77e9\u9635\u4e58\u6cd5\uff0c\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u63cf\u8ff0\u8fd9\u79cd\u6709\u5e8f\u3001\u975e\u4ea4\u6362\u4e16\u754c\u7684\u6570\u5b66\u8bed\u8a00\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>3. \u6027\u80fd\u4e0e\u5b9e\u73b0\uff1a\u8ba1\u7b97\u673a\u5982\u4f55\u9a6f\u670d\u8fd9\u5934\u5de8\u517d<\/strong><\/h2>\n\n\n\n<p>\u6211\u4eec\u5df2\u7ecf\u9886\u7565\u4e86\u77e9\u9635\u4e58\u6cd5\u7684\u7406\u8bba\u4f18\u96c5\u6027\uff0c\u4f46\u8981\u5728\u73b0\u5b9e\u4e16\u754c\u4e2d\u53d1\u6325\u5176\u5a01\u529b\uff0c\u6211\u4eec\u5fc5\u987b\u9762\u5bf9\u4e00\u4e2a\u6b8b\u9177\u7684\u5de5\u7a0b\u73b0\u5b9e\uff1a\u77e9\u9635\u4e58\u6cd5\u662f\u8ba1\u7b97\u5bc6\u96c6\u578b\u7684\u3002\u7406\u89e3\u8ba1\u7b97\u673a\u5982\u4f55\u9ad8\u6548\u5730\u6267\u884c\u8fd9\u4e00\u64cd\u4f5c\uff0c\u662f\u8fde\u63a5\u7406\u8bba\u4e0e\u5e94\u7528\uff0c\u5c24\u5176\u662f\u7406\u89e3\u73b0\u4ee3\u9ad8\u6027\u80fd\u8ba1\u7b97\u548c\u4eba\u5de5\u667a\u80fd\u786c\u4ef6\u7684\u5173\u952e\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u6734\u7d20\u7b97\u6cd5\u53ca\u5176\u6027\u80fd\u74f6\u9888<\/strong><\/h3>\n\n\n\n<p>\u8ba9\u6211\u4eec\u56de\u987e\u77e9\u9635\u4e58\u6cd5 <code>C = AB<\/code> \u7684\u5f62\u5f0f\u5316\u5b9a\u4e49\uff0c\u5176\u4e2d <code>A<\/code> \u662f <code>m x n<\/code> \u77e9\u9635, <code>B<\/code> \u662f <code>n x p<\/code> \u77e9\u9635, <code>C<\/code> \u662f <code>m x p<\/code> \u77e9\u9635\uff1a<br>$$C_{ij} = \\sum_{k=1}^{n} A_{ik} B_{kj}$$<br>\u8fd9\u4e2a\u5b9a\u4e49\u76f4\u63a5\u5c31\u53ef\u4ee5\u7ffb\u8bd1\u6210\u4e00\u4e2a\u4e09\u91cd\u5d4c\u5957\u5faa\u73af\u7684\u201c\u6734\u7d20\u201d\u7b97\u6cd5\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>for i from 0 to m-1:\n  for j from 0 to p-1:\n    sum = 0\n    for k from 0 to n-1:\n      sum += A&#91;i]&#91;k] * B&#91;k]&#91;j]\n    C&#91;i]&#91;j] = sum\nreturn C<\/code><\/pre>\n\n\n\n<p>\u8fd9\u4e2a\u7b97\u6cd5\u76f4\u89c2\u6613\u61c2\uff0c\u4f46\u5728\u6027\u80fd\u4e0a\u5374\u662f\u707e\u96be\u6027\u7684\u3002\u5bf9\u4e8e\u4e24\u4e2a <code>n x n<\/code> \u7684\u65b9\u9635\uff0c\u4e09\u4e2a\u5faa\u73af\u7684\u5c42\u6570\u5206\u522b\u662f <code>n<\/code>, <code>n<\/code>, <code>n<\/code>\u3002\u8fd9\u610f\u5473\u7740\u603b\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u64cd\u4f5c\u6b21\u6570\u4e0e <code>n<\/code> \u7684\u4e09\u6b21\u65b9\u6210\u6b63\u6bd4\u3002\u6211\u4eec\u79f0\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a <strong>O(n\u00b3)<\/strong>\u3002<\/p>\n\n\n\n<p><code>n<\/code> \u8f83\u5c0f\u65f6\uff0c\u8fd9\u4e0d\u6210\u95ee\u9898\u3002\u4f46\u5f53 <code>n<\/code> \u589e\u5927\u65f6\uff0c\u8ba1\u7b97\u91cf\u4f1a\u7206\u70b8\u6027\u589e\u957f\u3002\u4e00\u4e2a <code>1000 x 1000<\/code> \u7684\u77e9\u9635\uff0c\u5c31\u9700\u8981 <code>1000\u00b3 = 10\u4ebf<\/code>\u6b21\u4e58\u6cd5\u548c <code>10\u4ebf<\/code>\u6b21\u52a0\u6cd5\u3002\u5728\u4eca\u5929\u7684AI\u6a21\u578b\u4e2d\uff0c\u52a8\u8f84 <code>4096x4096<\/code> \u6216\u66f4\u5927\u7684\u77e9\u9635\u8fd0\u7b97\u53f8\u7a7a\u89c1\u60ef\uff0c<code>O(n\u00b3)<\/code> \u7684\u5929\u82b1\u677f\u5b9e\u5728\u592a\u4f4e\u4e86\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u8d85\u8d8a O(n\u00b3)\uff1a\u66f4\u5feb\u7684\u7406\u8bba\u7b97\u6cd5<\/strong><\/h3>\n\n\n\n<p>\u4e00\u4e2a\u81ea\u7136\u7684\u95ee\u9898\u662f\uff1a\u6211\u4eec\u80fd\u505a\u5f97\u6bd4 <code>O(n\u00b3)<\/code> \u66f4\u597d\u5417\uff1f<br>1969\u5e74\uff0c\u5fb7\u56fd\u6570\u5b66\u5bb6 Volker Strassen \u6295\u4e0b\u4e86\u4e00\u9897\u91cd\u78c5\u70b8\u5f39\u3002\u4ed6\u53d1\u73b0\uff0c\u5bf9\u4e8e <code>2x2<\/code> \u77e9\u9635\u4e58\u6cd5\uff0c\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e2a\u5de7\u5999\u7684\u6784\u9020\uff0c\u7528 7 \u6b21\u4e58\u6cd5 \u548c 18 \u6b21\u52a0\u51cf\u6cd5\u5b8c\u6210\uff0c\u800c\u4e0d\u662f\u6734\u7d20\u7b97\u6cd5\u6240\u9700\u7684 8 \u6b21\u4e58\u6cd5\u548c 4 \u6b21\u52a0\u6cd5\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='https:\/\/img.hcha.site\/images\/original\/portrait\/20250916_155206_4086.jpg'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  decoding=\"async\" data-original=\"https:\/\/img.hcha.site\/images\/original\/portrait\/20250916_155206_4086.jpg\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" style=\"width:250px;height:auto\"\/><\/div><figcaption class=\"wp-element-caption\"><strong>Volker Strassen<\/strong><br>1936 \u00b7 04 \u00b7 29 &#8211; \u81f3\u4eca<\/figcaption><\/figure>\n<\/div>\n\n\n<p>Strassen \u7b97\u6cd5\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5206\u6cbb\u6cd5\uff1a\u5c06\u5927\u77e9\u9635\u9012\u5f52\u5730\u62c6\u5206\u6210 <code>2x2<\/code> \u7684\u5b50\u77e9\u9635\u5757\uff0c\u7136\u540e\u7528\u4ed6\u7684 7 \u4e58\u6cd5\u516c\u5f0f\u6765\u8ba1\u7b97\u5757\u95f4\u7684\u4e58\u79ef\u3002\u867d\u7136\u52a0\u6cd5\u6b21\u6570\u589e\u591a\u4e86\uff0c\u4f46\u4e58\u6cd5\u662f\u66f4\u6602\u8d35\u7684\u64cd\u4f5c\uff0c\u4e14\u9012\u5f52\u7684\u5a01\u529b\u4f7f\u5f97\u6574\u4f53\u590d\u6742\u5ea6\u5f97\u4ee5\u964d\u4f4e\u3002\u5176\u6700\u7ec8\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a <strong>O(n^log\u2082(7))<\/strong>\uff0c\u7ea6\u7b49\u4e8e <strong>O(n^2.807)<\/strong>\u3002\u8fd9\u5728\u7406\u8bba\u4e0a\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u7a81\u7834\uff0c\u8bc1\u660e\u4e86 <code>O(n\u00b3)<\/code> \u5e76\u975e\u4e0d\u53ef\u903e\u8d8a\u7684\u94c1\u58c1\u3002<\/p>\n\n\n\n<p>\u5728 Strassen \u4e4b\u540e\uff0c\u4e00\u7cfb\u5217\u66f4\u590d\u6742\u7684\u7b97\u6cd5\u5728\u7406\u8bba\u4e0a\u5c06\u590d\u6742\u5ea6\u6307\u6570 <code>\u03c9<\/code> \u4e0d\u65ad\u63a8\u5411\u6781\u9650\u3002\u76ee\u524d\u7406\u8bba\u4e0a\u7684\u6700\u4f73\u6210\u7ee9 <code>\u03c9<\/code> \u5df2\u7ecf\u975e\u5e38\u63a5\u8fd1 2\u3002\u7136\u800c\uff0c\u8fd9\u4e9b\u7406\u8bba\u7b97\u6cd5\u7684\u5e38\u6570\u56e0\u5b50\u5de8\u5927\uff0c\u4e14\u5b9e\u73b0\u590d\u6742\uff0c\u5728\u5b9e\u8df5\u4e2d\u5f80\u5f80\u53ea\u5bf9\u6781\u5927\u5c3a\u5bf8\u7684\u77e9\u9635\u624d\u6709\u4f18\u52bf\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u771f\u5b9e\u4e16\u754c\u7684\u4f18\u5316\uff1a\u5185\u5b58\u5c42\u6b21\u4e0e\u5e76\u884c\u8ba1\u7b97<\/strong><\/h3>\n\n\n\n<p>\u5bf9\u4e8e\u5b9e\u9645\u7684\u8ba1\u7b97\u673a\u7cfb\u7edf\u800c\u8a00\uff0c<code>O(n\u00b3)<\/code> \u7684\u6734\u7d20\u7b97\u6cd5\u8fd8\u6709\u4e00\u4e2a\u66f4\u81f4\u547d\u7684\u95ee\u9898\uff1a<strong>\u7cdf\u7cd5\u7684\u5185\u5b58\u8bbf\u95ee\u6a21\u5f0f<\/strong>\u3002<\/p>\n\n\n\n<p>\u73b0\u4ee3CPU\u7684\u901f\u5ea6\u8fdc\u8d85\u5185\u5b58\u3002\u4e3a\u4e86\u5f25\u5408\u5dee\u8ddd\uff0cCPU\u5185\u7f6e\u4e86\u591a\u7ea7\u9ad8\u901f\u7f13\u5b58\uff08L1, L2, L3 \u7f13\u5b58\uff09\u3002\u5982\u679cCPU\u9700\u8981\u7684\u6570\u636e\u6070\u597d\u5728\u7f13\u5b58\u4e2d\uff0c\u901f\u5ea6\u6781\u5feb\uff1b\u5982\u679c\u4e0d\u5728\uff0c\u5c31\u5fc5\u987b\u53bb\u6162\u901f\u7684\u5185\u5b58\u4e2d\u8bfb\u53d6\uff0c\u5e26\u6765\u5de8\u5927\u7684\u5ef6\u8fdf\u60e9\u7f5a\u3002<\/p>\n\n\n\n<p>\u6734\u7d20\u7b97\u6cd5\u7684\u5185\u5b58\u8bbf\u95ee\u662f\u201c\u8df3\u8dc3\u5f0f\u201d\u7684\u3002\u5728\u8ba1\u7b97 <code>C[i][j]<\/code> \u65f6\uff0c\u5b83\u7ebf\u6027\u5730\u626b\u63cf <code>A<\/code> \u7684\u7b2c <code>i<\/code> \u884c\uff0c\u8fd9\u662f\u7f13\u5b58\u53cb\u597d\u7684\u3002\u4f46\u540c\u65f6\uff0c\u5b83\u5374\u8981\u201c\u5782\u76f4\u201d\u5730\u904d\u5386 <code>B<\/code> \u7684\u4e00\u6574\u5217\u3002\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u5b58\u5e03\u5c40\u4e2d\uff08\u884c\u4e3b\u5e8f\uff09\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u8bbf\u95ee <code>B<\/code> \u7684\u4e00\u4e2a\u5143\u7d20\uff0c\u90fd\u8981\u8df3\u8fc7\u4e00\u5927\u6bb5\u5185\u5b58\uff08\u4e00\u6574\u884c\u7684\u957f\u5ea6\uff09\uff0c\u6781\u6613\u5bfc\u81f4\u7f13\u5b58\u672a\u547d\u4e2d\u3002<\/p>\n\n\n\n<p>\u56e0\u6b64\uff0c\u73b0\u5b9e\u4e16\u754c\u4e2d\u9ad8\u6027\u80fd\u77e9\u9635\u4e58\u6cd5\u7684\u6838\u5fc3\uff0c\u5e76\u975e\u8ffd\u6c42\u7406\u8bba\u4e0a\u6700\u5feb\u7684\u7b97\u6cd5\uff0c\u800c\u662f<strong>\u6700\u5927\u5316\u7f13\u5b58\u5229\u7528\u7387<\/strong>\u548c<strong>\u5e76\u884c\u5316<\/strong>\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u5206\u5757<\/strong>: \u8fd9\u662f\u6700\u6838\u5fc3\u3001\u6700\u6709\u6548\u7684\u4f18\u5316\u3002\u5176\u601d\u60f3\u662f\u5c06\u5927\u77e9\u9635 <code>A<\/code> \u548c <code>B<\/code> \u5207\u5272\u6210\u5c0f\u7684\u5b50\u77e9\u9635\uff08\u5757\uff09\uff0c\u5927\u5c0f\u6b63\u597d\u80fd\u88c5\u5165CPU\u7684\u67d0\u4e00\u7ea7\u7f13\u5b58\u3002\u7136\u540e\uff0c\u7b97\u6cd5\u4e0d\u518d\u662f\u8ba1\u7b97\u5355\u4e2a\u5143\u7d20\u7684\u70b9\u79ef\uff0c\u800c\u662f\u5728\u8fd9\u4e9b\u5c0f\u5757\u4e4b\u95f4\u8fdb\u884c\u201c\u5757\u4e58\u6cd5\u201d\u3002\u901a\u8fc7\u7cbe\u5fc3\u8bbe\u8ba1\u7684\u5faa\u73af\u987a\u5e8f\uff0c\u53ef\u4ee5\u786e\u4fdd\u4e00\u65e6\u67d0\u51e0\u4e2a\u5757\u88ab\u52a0\u8f7d\u5230\u7f13\u5b58\u4e2d\uff0c\u5b83\u4eec\u4f1a\u88ab\u53cd\u590d\u5229\u7528\uff0c\u4ee5\u5b8c\u6210\u6240\u6709\u76f8\u5173\u7684\u5b50\u8ba1\u7b97\uff0c\u4ece\u800c\u5c06\u6602\u8d35\u7684\u5185\u5b58\u8bbf\u95ee\u6b21\u6570\u964d\u5230\u6700\u4f4e\u3002<\/li>\n\n\n\n<li><strong>\u5e76\u884c\u5316<\/strong>:\n<ul class=\"wp-block-list\">\n<li><strong>SIMD (Single Instruction, Multiple Data)<\/strong>\uff1a\u73b0\u4ee3CPU\u90fd\u652f\u6301SIMD\u6307\u4ee4\u96c6\uff08\u5982SSE, AVX\uff09\u3002\u8fd9\u4e9b\u6307\u4ee4\u5141\u8bb8\u4e00\u6761CPU\u6307\u4ee4\u540c\u65f6\u5bf9\u591a\u4e2a\u6570\u636e\uff08\u4f8b\u59824\u4e2a\u62168\u4e2a\u6d6e\u70b9\u6570\uff09\u6267\u884c\u76f8\u540c\u7684\u64cd\u4f5c\uff08\u5982\u4e58\u6cd5\u6216\u52a0\u6cd5\uff09\u3002\u77e9\u9635\u4e58\u6cd5\u4e2d\u7684\u70b9\u79ef\u8fd0\u7b97\uff0c\u5929\u7136\u9002\u5408\u7528SIMD\u6765\u52a0\u901f\u3002\u7f16\u8bd1\u5668\u548c\u5e95\u5c42\u5e93\u4f1a\u81ea\u52a8\u5c06\u5185\u5c42\u5faa\u73af\u201c\u5411\u91cf\u5316\u201d\uff0c\u4ee5\u5229\u7528\u8fd9\u79cd\u786c\u4ef6\u80fd\u529b\u3002<\/li>\n\n\n\n<li><strong>\u591a\u6838\u5e76\u884c<\/strong>: \u77e9\u9635\u4e58\u6cd5\u540c\u6837\u662f\u201c\u6613\u4e8e\u5e76\u884c\u201d\u7684\u3002\u7ed3\u679c\u77e9\u9635 <code>C<\/code> \u7684\u6bcf\u4e00\u4e2a\u5143\u7d20 <code>C[i,j]<\/code> \u7684\u8ba1\u7b97\u90fd\u662f\u72ec\u7acb\u7684\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u8f7b\u6613\u5730\u5c06 <code>C<\/code> \u7684\u884c\u6216\u5217\u7684\u8ba1\u7b97\u4efb\u52a1\uff0c\u5206\u914d\u7ed9\u4e0d\u540c\u7684CPU\u6838\u5fc3\u540c\u65f6\u6267\u884c\uff0c\u5b9e\u73b0\u63a5\u8fd1\u7ebf\u6027\u7684\u6027\u80fd\u63d0\u5347\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>GPU \u52a0\u901f<\/strong>: GPU \u5c06\u5e76\u884c\u8ba1\u7b97\u7684\u601d\u60f3\u63a8\u5411\u4e86\u6781\u81f4\u3002\u4e00\u4e2aGPU\u62e5\u6709\u6570\u5343\u4e2a\u5c0f\u578b\u7684\u3001\u76f8\u5bf9\u7b80\u5355\u7684\u8ba1\u7b97\u6838\u5fc3\u3002\u5b83\u7684\u67b6\u6784\u5929\u7136\u5c31\u662f\u4e3a\u5904\u7406\u5927\u89c4\u6a21\u3001\u5e76\u884c\u7684\u6d6e\u70b9\u8fd0\u7b97\u800c\u8bbe\u8ba1\u7684\u2014\u2014\u8fd9\u6b63\u662f\u77e9\u9635\u4e58\u6cd5\u7684\u672c\u8d28\u3002\u901a\u8fc7CUDA\uff08NVIDIA\uff09\u6216ROCm\uff08AMD\uff09\u7b49\u7f16\u7a0b\u6a21\u578b\uff0c\u53ef\u4ee5\u5c06\u5de8\u5927\u7684\u77e9\u9635\u4e58\u6cd5\u4efb\u52a1\u5206\u89e3\u6210\u6210\u5343\u4e0a\u4e07\u4e2a\u7ebf\u7a0b\uff0c\u6bcf\u4e2a\u7ebf\u7a0b\u8d1f\u8d23\u4e00\u5c0f\u90e8\u5206\u8ba1\u7b97\uff0c\u5728GPU\u7684\u201c\u8ba1\u7b97\u6d77\u6d0b\u201d\u4e2d\u540c\u65f6\u6267\u884c\u3002\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48GPU\u6210\u4e3a\u4e86\u6df1\u5ea6\u5b66\u4e60\u9769\u547d\u7684\u5f15\u64ce\u2014\u2014\u56e0\u4e3a\u6df1\u5ea6\u5b66\u4e60\u7684\u6838\u5fc3\uff0c\u5c31\u662f\u6d77\u91cf\u7684\u77e9\u9635\u4e58\u6cd5\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u5728\u5b9e\u8df5\u4e2d\uff0c\u6ca1\u6709\u4eba\u4f1a\u624b\u5199\u7528\u4e8e\u751f\u4ea7\u73af\u5883\u7684\u77e9\u9635\u4e58\u6cd5\u5faa\u73af\u3002\u5927\u5bb6\u90fd\u4f1a\u4f9d\u8d56\u9ad8\u5ea6\u4f18\u5316\u7684\u5e95\u5c42\u6570\u5b66\u5e93\uff0c\u5982 BLAS (Basic Linear Algebra Subprograms)\u3002\u50cf OpenBLAS, Intel MKL, Apple Accelerate \u7b49\u5e93\uff0c\u5b83\u4eec\u7684\u4f8b\u7a0b\uff0c\u662f\u7531\u9876\u7ea7\u7684\u7f16\u8bd1\u5668\u4e13\u5bb6\u548c\u786c\u4ef6\u5de5\u7a0b\u5e08\uff0c\u7ed3\u5408\u4e86\u5206\u5757\u3001SIMD\u3001\u591a\u7ebf\u7a0b\u7b49\u6240\u6709\u4f18\u5316\u624b\u6bb5\uff0c\u5e76\u9488\u5bf9\u7279\u5b9aCPU\u67b6\u6784\u624b\u5199\u6c47\u7f16\u8c03\u4f18\u7684\u7cbe\u534e\u4e4b\u4f5c\u3002\u5f53\u6211\u4eec\u5728 Python \u4e2d\u8c03\u7528 NumPy \u7684 <code>np.dot(A, B)<\/code> \u65f6\uff0c\u80cc\u540e\u6b63\u662f\u8fd9\u4e9b\u201c\u6027\u80fd\u731b\u517d\u201d\u5728\u65e0\u58f0\u5730\u5486\u54ee\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>4. \u591a\u7ef4\u5ea6\u5206\u6790\uff1a\u8d85\u8d8a\u8ba1\u7b97\u7684\u610f\u4e49<\/strong><\/h2>\n\n\n\n<p>\u77e9\u9635\u4e58\u6cd5\u7684\u5f71\u54cd\u529b\u8fdc\u8fdc\u8d85\u51fa\u4e86\u7eaf\u6570\u5b66\u548c\u9ad8\u6027\u80fd\u8ba1\u7b97\u7684\u8303\u7574\u3002\u5b83\u662f\u4e00\u79cd\u901a\u7528\u8bed\u8a00\uff0c\u88ab\u7528\u6765\u63cf\u8ff0\u548c\u64cd\u7eb5\u5404\u4e2a\u79d1\u5b66\u4e0e\u5de5\u7a0b\u9886\u57df\u7684\u7cfb\u7edf\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66<\/strong>: \u57283D\u6e38\u620f\u4e2d\u4f60\u770b\u5230\u7684\u6bcf\u4e00\u4e2a\u89d2\u8272\u7684\u79fb\u52a8\u3001\u65cb\u8f6c\u548c\u7f29\u653e\uff0c\u80cc\u540e\u90fd\u662f\u77e9\u9635\u4e58\u6cd5\u5728\u9a71\u52a8\u3002\u4e00\u4e2a3D\u6a21\u578b\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u53ef\u4ee5\u7528\u4e00\u4e2a\u5411\u91cf <code>(x, y, z)<\/code> \u8868\u793a\u3002\u4e00\u4e2a\u53d8\u6362\u77e9\u9635\uff08\u5982\u65cb\u8f6c\u77e9\u9635\u3001\u7f29\u653e\u77e9\u9635\u3001\u5e73\u79fb\u77e9\u9635\uff09\u4e58\u4ee5\u8fd9\u4e2a\u9876\u70b9\u5411\u91cf\uff0c\u5c31\u4f1a\u5f97\u5230\u9876\u70b9\u5728\u53d8\u6362\u540e\u7684\u65b0\u4f4d\u7f6e\u3002\u5c06\u4e00\u7cfb\u5217\u53d8\u6362\u7ec4\u5408\u8d77\u6765\uff0c\u5c31\u5bf9\u5e94\u7740\u4e00\u7cfb\u5217\u53d8\u6362\u77e9\u9635\u7684\u8fde\u4e58\u3002\u8fd9\u6837\u7684\u6d41\u6c34\u7ebf\uff0c\u662f\u5b9e\u65f6\u6e32\u67d3\u7684\u6839\u57fa\u3002<\/li>\n\n\n\n<li><strong>\u4eba\u5de5\u667a\u80fd\u4e0e\u673a\u5668\u5b66\u4e60<\/strong>: \u77e9\u9635\u4e58\u6cd5\u662f\u795e\u7ecf\u7f51\u7edc\u7684\u5fc3\u8df3\u3002\u4e00\u4e2a\u6700\u57fa\u672c\u7684\u524d\u9988\u795e\u7ecf\u7f51\u7edc\u5c42\u53ef\u4ee5\u8868\u793a\u4e3a <code>\u8f93\u51fa = \u6fc0\u6d3b\u51fd\u6570(W * \u8f93\u5165 + b)<\/code>\u3002\u5176\u4e2d\uff0c<code>\u8f93\u5165<\/code> \u662f\u4e00\u4e2a\u5305\u542b\u8f93\u5165\u7279\u5f81\u7684\u5411\u91cf\uff0c<code>W<\/code> \u662f\u4e00\u4e2a\u6743\u91cd\u77e9\u9635\uff0c<code>b<\/code> \u662f\u504f\u7f6e\u5411\u91cf\u3002<code>W * \u8f93\u5165<\/code> \u8fd9\u4e2a\u6838\u5fc3\u7684\u77e9\u9635-\u5411\u91cf\u4e58\u6cd5\uff0c\u8ba1\u7b97\u4e86\u6240\u6709\u8f93\u5165\u7279\u5f81\u7684\u52a0\u6743\u548c\u3002\u6743\u91cd\u77e9\u9635 <code>W<\/code> \u7684 <code>[i, j]<\/code> \u5143\u7d20\uff0c\u4ee3\u8868\u4e86\u7b2c <code>j<\/code> \u4e2a\u8f93\u5165\u5bf9\u7b2c <code>i<\/code> \u4e2a\u8f93\u51fa\u795e\u7ecf\u5143\u7684\u5f71\u54cd\u529b\u3002\u6a21\u578b\u7684\u201c\u5b66\u4e60\u201d\u8fc7\u7a0b\uff0c\u672c\u8d28\u4e0a\u5c31\u662f\u901a\u8fc7\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\uff0c\u4e0d\u65ad\u5fae\u8c03\u8fd9\u4e2a\u5de8\u5927\u7684\u6743\u91cd\u77e9\u9635 <code>W<\/code> \u7684\u8fc7\u7a0b\u3002\u50cfGPT-4\u8fd9\u6837\u7684\u5927\u8bed\u8a00\u6a21\u578b\uff0c\u5176\u5185\u90e8\u8fdb\u884c\u7684\u8fd0\u7b97\uff0c\u7edd\u5927\u90e8\u5206\u90fd\u53ef\u4ee5\u5f52\u7ed3\u4e3a\u5de8\u578b\u77e9\u9635\u4e4b\u95f4\u7684\u4e58\u6cd5\u3002<\/li>\n\n\n\n<li><strong>\u91cf\u5b50\u8ba1\u7b97<\/strong>: \u5728\u91cf\u5b50\u4e16\u754c\u4e2d\uff0c\u4e00\u4e2a\u91cf\u5b50\u6bd4\u7279\u7684\u72b6\u6001\u53ef\u4ee5\u7528\u4e00\u4e2a\u590d\u6570\u5411\u91cf\u6765\u63cf\u8ff0\u3002\u5bf9\u91cf\u5b50\u6bd4\u7279\u8fdb\u884c\u7684\u64cd\u4f5c\uff0c\u5373\u91cf\u5b50\u95e8\uff0c\u5219\u7531\u9149\u77e9\u9635\u6765\u8868\u793a\u3002\u91cf\u5b50\u7b97\u6cd5\u7684\u6267\u884c\u8fc7\u7a0b\uff0c\u5c31\u662f\u7528\u4ee3\u8868\u91cf\u5b50\u95e8\u7684\u77e9\u9635\uff0c\u53bb\u4e58\u4ee5\u4ee3\u8868\u91cf\u5b50\u7cfb\u7edf\u5f53\u524d\u72b6\u6001\u7684\u5411\u91cf\uff0c\u4ece\u800c\u6f14\u5316\u51fa\u65b0\u7684\u72b6\u6001\u3002\u77e9\u9635\u4e58\u6cd5\u662f\u6a21\u62df\u548c\u7406\u89e3\u91cf\u5b50\u7b97\u6cd5\u7684\u6570\u5b66\u57fa\u7840\u3002<\/li>\n\n\n\n<li><strong>\u56fe\u8bba<\/strong>: \u4e00\u4e2a\u56fe\u53ef\u4ee5\u7528\u90bb\u63a5\u77e9\u9635 <code>A<\/code> \u6765\u8868\u793a\u3002\u5982\u679c\u9876\u70b9<code>i<\/code>\u548c<code>j<\/code>\u4e4b\u95f4\u6709\u8fb9\u76f8\u8fde\uff0c\u5219<code>A[i,j] = 1<\/code>\uff0c\u5426\u5219\u4e3a0\u3002\u77e9\u9635\u4e58\u6cd5\u7684\u5a01\u529b\u5728\u8fd9\u91cc\u5f97\u5230\u4e86\u60ca\u4eba\u7684\u4f53\u73b0\uff1a<code>A\u00b2 = A * A<\/code> \u8fd9\u4e2a\u65b0\u77e9\u9635\uff0c\u5176\u5143\u7d20 <code>(A\u00b2)[i,j]<\/code> \u7684\u503c\uff0c\u6070\u597d\u662f\u4ece\u9876\u70b9 <code>i<\/code> \u5230\u9876\u70b9 <code>j<\/code> \u957f\u5ea6\u4e3a <strong>2<\/strong> \u7684\u8def\u5f84\u6570\u91cf\uff01\u66f4\u4e00\u822c\u5730\uff0c<code>(A^k)[i,j]<\/code> \u7ed9\u51fa\u4e86\u4ece <code>i<\/code> \u5230 <code>j<\/code> \u957f\u5ea6\u4e3a <code>k<\/code> \u7684\u8def\u5f84\u6570\u91cf\u3002<\/li>\n\n\n\n<li><strong>\u5b89\u5168\u6027\u5206\u6790<\/strong>: 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class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 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