{"id":19788,"date":"2026-01-25T10:47:38","date_gmt":"2026-01-25T02:47:38","guid":{"rendered":"https:\/\/www.orczhou.com\/?p=19788"},"modified":"2026-02-01T19:23:52","modified_gmt":"2026-02-01T11:23:52","slug":"my-linear-algebra-learning-01","status":"publish","type":"post","link":"https:\/\/www.orczhou.com\/index.php\/2026\/01\/my-linear-algebra-learning-01\/","title":{"rendered":"\u7ebf\u6027\u4ee3\u6570\u56de\u987e"},"content":{"rendered":"\n\n\n\n<p style=\"margin-top:4px\">\u5982\u679c\u8981\u7406\u89e3\u5927\u6a21\u578b\u5185\u90e8\u662f\u5982\u4f55\u5de5\u4f5c\u7684\uff0c\u826f\u597d\u7684\u6570\u5b66\u57fa\u7840\u662f\u5fc5\u987b\u7684\uff0c\u800c\u7ebf\u6027\u4ee3\u6570\u53c8\u662f\u6240\u6709\u8fd9\u4e9b\u7684\u57fa\u7840\u3002\u4f8b\u5982\u6211\u4eec\u6765\u770b \\(\\text{Attention} \\) \u673a\u5236\u4e2d\u7684\u5982\u4e0b\u95ee\u9898\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u4e3a\u4ec0\u4e48\u8981\u91cd\u6e29\u7ebf\u6027\u4ee3\u6570<\/h3>\n\n\n\n<p>\u8003\u8651\u7b2c \\(j \\) \u4e2a \\( \\text{Layer} \\) \u4e2d\u7684\u7b2c \\( i \\) \u4e2a \\( \\text{Head} \\) \uff0c\u5219\u6709\u5982\u4e0b\u7684\u8ba1\u7b97\uff08\u4e3a\u4e86\u7b80\u5316\uff0c\u4e0b\u8ff0\u7684\u89d2\u6807\u7701\u7565\u4e86 \\( \\text{Layer} \\) \u90e8\u5206\uff09\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\nQ_i &#038;= XW_i^Q \\\\[0.3em]<br \/>\nK_i &#038;= XW_i^K \\\\[0.3em]<br \/>\n\\text{Therefore:} \\\\[0.3em]<br \/>\n\\text{Attention Score}_i &#038;= Q_iK_i^T \\\\[0.3em]<br \/>\n&#038;= XW_i^Q(XW_i^K)^T \\\\[0.3em]<br \/>\n&#038;= XW_i^Q\\big((W_i^K)^T X^T\\big) \\\\[0.3em]<br \/>\n&#038;= XW_i^Q(W_i^K)^T X^T \\\\[0.3em]<br \/>\n&#038;= X\\big(W_i^Q (W_i^K)^T\\big) X^T \\\\[0.3em]<br \/>\n\\text{Let:} \\quad \\\\[0.3em]<br \/>\nW_i^{QK} &#038;= W_i^Q (W_i^K)^T \\\\[0.3em]<br \/>\n\\text{Therefore:} \\\\[0.3em]<br \/>\n\\text{Attention Score}_i &#038;= XW_i^{QK} X^T \\\\[0.3em]<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u6ce8\uff1a\u4e0a\u8ff0\u7684\u63a8\u5bfc\u4f7f\u7528\u77e9\u9635\u7684\u4e00\u4e9b\u7b80\u5355\u7279\u6027\uff0c\u5305\u62ec\u8f6c\u7f6e\u8ba1\u7b97\u3001\u7ed3\u5408\u5f8b\u7b49\u3002\u5176\u4e2d\uff0c\u5728\u5f00\u6e90\u7684GPT2\u6a21\u578b\u4e2d\uff0c\\(W_i^Q \\,, W_i^K \\) \u90fd\u662f \\(64 \\times 768 \\)\u7684\u77e9\u9635\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc\u4e00\u4e2a\u7b80\u5355\u3001\u53c8\u4e0d\u592a\u7b80\u5355\u7684\u95ee\u9898\u662f\uff1a\u90a3\u4e3a\u4ec0\u4e48\u6bcf\u4e00\u4e2a \\( \\text{Head} \\) \u4e2d\u4e0d\u4f7f\u7528\u4e00\u4e2a\u6743\u91cd \\( W_i^{QK} \\) \u5c31\u53ef\u4ee5\u4e86\uff1f\u8fd9\u4e2a\u95ee\u9898\u4ece\u6211\u7b2c\u4e00\u6b21\u770b\u660e\u767d \\( \\text{Attention} \\) \u7684\u8ba1\u7b97\u540e\uff0c\u56f0\u6270\u4e86\u6211\u4e00\u4f1a\u513f\uff0c\u76f4\u5230\u91cd\u6e29\u4e86\u7ebf\u6027\u4ee3\u6570\uff0c\u624d\u7b97\u662f\u7406\u89e3\u4e86\u4e0a\u8ff0\u7684\u8ba1\u7b97\u3002<\/p>\n\n\n\n<p>\u5927\u5b66\u65f6\u5b66\u4e60\u7ebf\u6027\u4ee3\u6570\u5b66\u5f97\u975e\u5e38\u75db\u82e6\uff0c\u800c\u73b0\u5728\u5e26\u7740\u95ee\u9898\u518d\u53bb\u770b\u8fd9\u672c\u4e66\uff0c\u7adf\u7136\u82b1\u4e86\u4e24\u4e2a\u665a\u4e0a\u5c31\u770b\u5b8c\u4e86\u3002\u8fd9\u91cc\u5bf9\u91cc\u9762\u7684\u57fa\u672c\u6982\u5ff5\u548c\u7ed3\u8bba\u505a\u4e2a\u68b3\u7406\uff0c\u4ee5\u66f4\u597d\u7684\u7406\u89e3\u4ec0\u4e48\u662f\u201c\u7ebf\u6027\u53d8\u6362\u201d\u3001\u4e0e\u77e9\u9635\u7684\u5173\u7cfb\u662f\u4ec0\u4e48\u3001\u5982\u4f55\u7814\u7a76\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u6216\u77e9\u9635\u7684\u6027\u8d28\u7b49\u3002<\/p>\n\n\n\n<p>\u672c\u7cfb\u5217\u4e3b\u8981\u4ee5\u4ecb\u7ecd\u7ebf\u6027\u4ee3\u6570\u7684\u201c\u76f4\u89c9\u201d\u4e3a\u4e3b\uff0c\u4e0d\u4f1a\u505a\u4efb\u4f55\u8bc1\u660e\uff0c\u4e3a\u4e86\u66f4\u597d\u7684\u9610\u8ff0\u201c\u76f4\u89c9\u201d\uff0c\u751a\u81f3\u727a\u7272\u4e86\u5f88\u591a\u7684\u6570\u5b66\u4e25\u8c28\uff0c\u770b\u5ba2\u4e5f\u9700\u8981\u4ece\u6784\u5efa\u201c\u76f4\u89c9\u201d\u4e0e\u201c\u8054\u7cfb\u201d\u7684\u89d2\u5ea6\u9605\u8bfb\u3002\u672c\u6587\u7684\u9605\u8bfb\u524d\u63d0\u662f\u5df2\u7ecf\u6709\u4e00\u5b9a\u7684\u7ebf\u6027\u4ee3\u6570\u7684\u57fa\u7840\u3001\u4e5f\u5bf9\u795e\u7ecf\u7f51\u7edc\/LLM\u6280\u672f\u6709\u4e00\u5b9a\u4e86\u89e3\uff0c\u90a3\u4e48\u672c\u6587\u5219\u5c1d\u8bd5\u901a\u8fc7\u8f83\u5c0f\u7684\u7bc7\u5e45\u53bb\u6784\u5efa\u4e24\u8005\u7684\u8054\u7cfb\uff0c\u770b\u770b\u5982\u4f55\u4f7f\u7528\u7ebf\u6027\u4ee3\u6570\u7684\u6280\u672f\u53bb\u7814\u7a76\u795e\u7ecf\u7f51\u7edc\u7684\u4e2d\u7684\u95ee\u9898\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u7ebf\u6027\u4ee3\u6570\u8ba8\u8bba\u7684\u4e3b\u8981\u95ee\u9898<\/h3>\n\n\n\n<p>\u901a\u5e38\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u8bfe\u7a0b\u4f1a\u4ece \\(n\\) \u5143\u4e00\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u5f15\u5165\uff0c\u5e76\u4f7f\u7528\u884c\u5217\u5f0f\u7406\u8bba\uff0c\u53bb\u8f83\u4e3a\u5f7b\u5e95\u7684\u56de\u7b54\u5982\u4f55\u89e3\u51b3 \\(n\\) \u5143\u4e00\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u66f4\u8fdb\u4e00\u6b65\u7684\uff0c\u4e3a\u4e86\u66f4\u597d\/\u66f4\u5b8c\u6574\u7684\u7814\u7a76 \\(n\\) \u5143\u4e00\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u201c\u89e3\u7a7a\u95f4\u201d\uff0c\u5219\u9700\u8981\u5f15\u5165\u4e00\u4e2a\u65b0\u7684\u7814\u7a76\u5bf9\u8c61\uff1a\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u3002\u800c\u5411\u91cf\u7a7a\u95f4\u672c\u8eab\u6240\u5177\u5907\u7684\u666e\u904d\u6027\uff0c\u5df2\u7ecf\u8fdc\u8d85\u51fa \\(n\\) \u5143\u4e00\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u672c\u8eab\u3002\u800c\u540e\uff0c\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u3001\u201c\u7ebf\u6027\u53d8\u6362\u201d\u5c31\u53d8\u6210\u4e86\u65b0\u7684\u7814\u7a76\u5bf9\u8c61\uff0c\u56e0\u4e3a\u73b0\u5b9e\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u7ecf\u5e38\u4f1a\u5c1d\u8bd5\u901a\u8fc7\u201c\u7ebf\u6027\u53d8\u6362\u201d\u6765\u6d1e\u5bdf\u5411\u91cf\u7a7a\u95f4\u4e2d\u5411\u91cf\u7684\u5173\u7cfb\u3002<\/p>\n\n\n\n<p>\u662f\u4e0d\u662f\u611f\u89c9\u4e0a\u9762\u63cf\u8ff0\u6f0f\u4e86\u4ec0\u4e48\uff1f\u662f\u7684\uff0c\u6f0f\u4e86\u201c\u77e9\u9635\u201d\u3002\u65e0\u8bba\u662f\u8ba8\u8bba \\(n\\) \u5143\u4e00\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u8fd8\u662f\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u6216\u201c\u7ebf\u6027\u53d8\u6362\u201d\uff0c\u77e9\u9635\u90fd\u662f\u201c\u6838\u5fc3\u201d\u5de5\u5177\u3002\u8fd9\u4e2a\u5de5\u5177\u201c\u6838\u5fc3\u201d\u6216\u8005\u8bf4\u91cd\u8981\u5230\u4ec0\u4e48\u7a0b\u5ea6\u5462\uff1f\u751a\u81f3\u5f88\u591a\u95ee\u9898\uff0c\u53ea\u9700\u8981\u7814\u7a76\u6e05\u695a\u5bf9\u5e94\u201c\u77e9\u9635\u201d\u7684\u7279\u6027\uff0c\u539f\u6765\u7684\u95ee\u9898\u5c31\u7814\u7a76\u6e05\u695a\u4e86\u3002\u6240\u4ee5\uff0c\u4f60\u4f1a\u6ce8\u610f\u5230\uff0c\u7ebf\u6027\u4ee3\u6570\u7684\u4e66\u4e2d\uff0c\u51e0\u4e4e\u5168\u90fd\u5728\u4ecb\u7ecd\u201c\u77e9\u9635\u201d\u7684\u5404\u79cd\u7279\u6027\u3002<\/p>\n\n\n\n<p>\u800c\u5404\u79cd \\( \\text{Embedding} \\) \u5c31\u53ef\u4ee5\u7406\u89e3\u4e3a\u662f\u5728\u6700\u4e3a\u5178\u578b\u7684\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. \\(n \\) \u5143\u4e00\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3<\/h3>\n\n\n\n<p>\u7ebf\u6027\u4ee3\u6570\u901a\u5e38\u90fd\u4f1a\u4ee5\u89e3\u201c\\( n \\)\u5143\u4e00\u6b21\u65b9\u7a0b\u7ec4\u201d\u4e3a\u5207\u5165\u70b9\uff0c\u8fd9\u4e2a\u95ee\u9898\u770b\u4f3c\u5f88\u7b80\u5355\uff0c\u4f46\u662f\u6700\u7ec8\u8981\u5b8c\u5168\u8ba8\u8bba\u6e05\u695a\uff0c\u5219\u9700\u8981\u5f88\u591a\u7bc7\u5e45\u3002\u8fd9\u4e5f\u662f\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u524d\u534a\u90e8\u5206\u6bd4\u8f83\u67af\u71e5\u7684\u539f\u56e0\u3002\u6574\u4f53\u4e0a\u6765\u770b\uff0c\u5173\u4e8e\u201c\\( n \\)\u5143\u4e00\u6b21\u65b9\u7a0b\u7ec4\u201d\u7684\u89e3\u9700\u8981\u8ba8\u8bba\u6e05\u695a\u51e0\u4e2a\u95ee\u9898\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list is-style-no-disc\">\n<li>(a) \\( n \\)\u5143\u4e00\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3\u662f\u5426\u5b58\u5728\uff1f <\/li>\n\n\n\n<li>(b) \u5982\u679c\u5b58\u5728\uff0c\u5982\u4f55\u6c42\u89e3 <\/li>\n\n\n\n<li>(c) \u5982\u679c\u89e3\u4e0d\u5b58\u5728\uff0c\u5145\u8981\u6761\u4ef6\u662f\u4ec0\u4e48 <\/li>\n\n\n\n<li>(d) \u5982\u679c\u6709\u89e3\uff0c\u90a3\u4e48\u6240\u6709\u7684\u89e3\u5982\u4f55\u8868\u8fbe<\/li>\n<\/ul>\n\n\n\n<p>\u5728\u63a2\u8ba8\u4e0a\u8ff0\u95ee\u9898\u7684\u65f6\u5019\uff0c\u5148\u662f\u5f15\u5165\u4e86\u201c\u884c\u5217\u5f0f\u201d\u3001\u201c\u77e9\u9635\u201d\u7684\u6982\u5ff5\u4e0e\u7406\u8bba\uff0c\u5e76\u901a\u8fc7\u77e9\u9635\u7684\u201c\u521d\u7b49\u53d8\u6362\u201d\u5b9e\u73b0\u5bf9\u4e0a\u8ff0\u95ee\u9898\u7684\u6c42\u89e3\u3002\u8fd9\u91cc\u6d89\u53ca\u5230\u7684\u6982\u5ff5\u5ef6\u4f38\u51fa\u4e86\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list is-style-no-disc\">\n<li>(a) \u77e9\u9635\u7684\u521d\u7b49\u53d8\u6362 <\/li>\n\n\n\n<li>(b) \u77e9\u9635\u7684\u79e9\u7b49<\/li>\n<\/ul>\n\n\n\n<p>\u8fd9\u91cc\u6211\u4eec\u5217\u4e3e\u4e00\u4e9b\u4e3b\u8981\u7684\u7ed3\u8bba\uff08\u5e76\u4e0d\u505a\u63a8\u5012\uff0c\u63a8\u5012\u8fc7\u7a0b\u8fd8\u662f\u975e\u5e38\u590d\u6742\u7684\uff0c\u8fd9\u4e5f\u6b63\u662f\u7ebf\u6027\u4ee3\u6570\u4e66\u6bd4\u8f83\u67af\u71e5\u7684\u539f\u56e0\u4e4b\u4e00\uff09\u3002\u8fd9\u91cc\u8003\u8651\u5982\u4e0b\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff1a<\/p>\n\n\n<p>$$<br \/>\na_{11}x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n = b_1, \\\\<br \/>\na_{21}x_1 + a_{22}x_2 + \\cdots + a_{2n}x_n = b_2, \\\\<br \/>\n\\quad\\vdots \\\\<br \/>\na_{m1}x_1 + a_{m2}x_2 + \\cdots + a_{mn}x_n = b_m.<br \/>\n$$<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark> \u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u89e3\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\uff1a\u5b83\u7684\u7cfb\u6570\u77e9\u9635\u4e0e\u589e\u5e7f\u77e9\u9635\u6709\u76f8\u540c\u7684\u79e9\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark> \u7ebf\u6027\u65b9\u7a0b\u7ec4\u7cfb\u6570\u77e9\u9635\u548c\u589e\u5e7f\u77e9\u9635\u7684\u79e9\u90fd\u662f\\( r \\)\uff0c\u65b9\u7a0b\u7ec4\u7684\u672a\u77e5\u6570\u7684\u4e2a\u6570\u662f\\( n \\)\uff0c\u5982\u679c\uff1a<\/p>\n\n\n\n<ul style=\"padding-right:var(--wp--preset--spacing--small);padding-left:var(--wp--preset--spacing--small)\" class=\"wp-block-list\">\n<li>\\( r = n \\) \u5219\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u89e3<\/li>\n\n\n\n<li>\\( r &lt; n \\) \u5219\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u65e0\u7a77\u7ec4\u89e3<\/li>\n<\/ul>\n\n\n\n<p>\u4e0a\u8ff0\u4e24\u4e2a\u5b9a\u7406\u8f83\u4e3a\u5f7b\u5e95\u7684\u56de\u7b54\u4e86\u89e3\u5b58\u5728\u6027\u7684\u95ee\u9898\u3002\u90a3\u4e48\uff0c\u89e3\u7684\u516c\u5f0f\u5316\u8868\u8fbe\u662f\u600e\u6837\u7684\u5462\uff1f\u4e3a\u4e86\u7565\u5fae\u7b80\u5316\u95ee\u9898\uff0c\u8fd9\u91cc\u8003\u8651\u4ec5\u8003\u8651\\( n \\)\u4e2a\u65b9\u7a0b\u3001\\( n \\)\u4e2a\u672a\u77e5\u6570\uff0c\u4e14\u89e3\u552f\u4e00\u7684\u60c5\u51b5\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{cases}<br \/>\na_{11}x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n = b_1, \\\\<br \/>\na_{21}x_1 + a_{22}x_2 + \\cdots + a_{2n}x_n = b_2, \\\\<br \/>\n\\quad\\vdots \\\\<br \/>\na_{n1}x_1 + a_{n2}x_2 + \\cdots + a_{nn}x_n = b_n<br \/>\n\\end{cases}<br \/>\n\\quad\\Longleftrightarrow\\quad<br \/>\n\\begin{bmatrix}<br \/>\na_{11} &#038; a_{12} &#038; \\cdots &#038; a_{1n} \\\\<br \/>\na_{21} &#038; a_{22} &#038; \\cdots &#038; a_{2n} \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\na_{n1} &#038; a_{n2} &#038; \\cdots &#038; a_{nn}<br \/>\n\\end{bmatrix}<br \/>\n\\begin{bmatrix}<br \/>\nx_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\nb_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n<br \/>\n\\end{bmatrix}<br \/>\n$$<\/p>\n\n\n\n<p>\u5f62\u5f0f\u5316\u7684\u89e3\uff0c\u5219\u53ef\u4ee5\u7531\u4e24\u79cd\u65b9\u5f0f\u7ed9\u51fa\uff1a<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u4f7f\u7528\u77e9\u9635\u7684\u8868\u8fbe\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\vec{x} = A^{-1}\\vec{b}<br \/>\n$$<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>\u201c<a href=\"https:\/\/zh.wikipedia.org\/zh-cn\/%E5%85%8B%E8%90%8A%E5%A7%86%E6%B3%95%E5%89%87\">\u514b\u83b1\u9ed8\u6cd5\u5219<\/a>\u201d\uff08Cramer\u2019s rule \/ formula\uff09\uff1a<\/p>\n\n\n<p>$$<br \/>\nx_1 = \\frac{D_1}{D},x_2 = \\frac{D_2}{D},\\dots , x_n = \\frac{D_n}{D},<br \/>\n$$<\/p>\n<\/div>\n<\/div>\n\n\n\n<p>\u6ce8\u610f\uff1a\u4e0a\u8ff0\u7684\u4e24\u79cd\u8868\u8fbe\uff0c\u65e0\u8bba\u54ea\u79cd\uff0c\u9650\u5236\u6761\u4ef6\u90fd\u975e\u5e38\u82db\u523b\uff0c\u5373\u8981\u6c42\uff1a\u77e9\u9635\\( A \\)\u53ef\u9006\u6216\u8005\u884c\u5217\u5217\u5f0f\\(D \\neq 0\\)\uff0c\u5f53\u7136\uff0c\u8fd9\u4e24\u4e2a\u6761\u4ef6\u5f0f\u7b49\u4ef7\u7684\u3002\u5e76\u4e14\uff0c\u8fd9\u91cc\u7684\\(D\\)\u4e5f\u7ecf\u5e38\u5199\u4f5c\uff1a\\( det(A) \\)\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. \u77e9\u9635\u57fa\u7840\u4e0e\u90e8\u5206\u7ed3\u8bba<\/h3>\n\n\n\n<p>\u867d\u7136\u662f\u4e3a\u4e86\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u624d\u5f15\u5165\u77e9\u9635\u7684\uff0c\u4f46\u5f88\u5feb\u5c31\u4f1a\u610f\u8bc6\u5230\uff0c\u5bf9\u77e9\u9635\u672c\u8eab\u7279\u6027\u7684\u7814\u7a76\u6709\u7740\u66f4\u4e3a\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n<p>\u9996\u5148\uff0c\u5728\u5b9a\u4e49\u4e86\u77e9\u9635\u7684\u8fd0\u7b97\u4e4b\u540e\uff0c\u5f88\u5feb\u5c31\u4f1a\u6709\u4e00\u4e9b\u77e9\u9635\u7684\u8fd0\u7b97\u5f8b\u3002\u4f8b\u5982\uff0c\u52a0\u6cd5\u7684\u7ed3\u5408\u5f8b\u3001\u4ea4\u6362\u5f8b\u3001\u5206\u914d\u5f8b\u7b49\u3002\u8fd9\u91cc\uff0c\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u91cd\u70b9\uff0c\u4e14\u7565\u5fae\u590d\u6742\u4e00\u4e9b\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u9996\u5148\uff0c\u9996\u5148\u77e9\u9635\u7684\u4e58\u6cd5\u662f\u4e0d\u6ee1\u8db3\u4ea4\u6362\u5f8b\u7684<\/li>\n\n\n\n<li>\u5f88\u5e78\u8fd0\uff0c\u7ed3\u5408\u5f8b\u548c\u5206\u914d\u5f8b\u90fd\u662f\u6ee1\u8db3\u7684<\/li>\n\n\n\n<li>\u518d\u6b21\uff0c\u5bf9\u4e8e\u8f6c\u7f6e\u8fd0\u7b97\uff0c\u662f\u6ee1\u8db3\u5982\u4e0b\u5f62\u5f0f\u7684\uff1a\\( (AB)^T = B^TA^T \\)<\/li>\n<\/ul>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark> \u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u521d\u7b49\u53d8\u6362\uff0c\u5bf9\u5e94\u7740\u4e09\u4e2a\u521d\u7b49\u53d8\u6362\u77e9\u9635\uff1a\\( P_{ij} \\)\u3001\\( D_i(k) \\)\u3001\\( T_{ij}(k) \\)\uff0c\u4e14\u8fd9\u4e09\u4e2a\u521d\u7b49\u53d8\u6362\u77e9\u9635\u90fd\u662f\u53ef\u9006\u7684\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark> \u4e00\u4e2a\\( m \\times n\\)\u7684\u77e9\u9635\\( A \\)\u603b\u662f\u53ef\u4ee5\u901a\u8fc7\u521d\u7b49\u53d8\u6362\u5316\u4e3a\u4ee5\u4e0b\u5f62\u5f0f\u7684\u77e9\u9635\uff1a<\/p>\n\n\n<p>$$ \\bar{A} = \\begin{bmatrix}<br \/>\nI_r &#038; O_{r,\\,n-r} \\\\<br \/>\nO_{m-r,\\,r} &#038; O_{m-r,\\,n-r}<br \/>\n\\end{bmatrix} $$<\/p>\n\n\n\n<p>\u8fd9\u91cc\uff0c\\(I_r\\)\u662f\\( r \\)\u9636\u5355\u4f4d\u77e9\u9635\uff0c\\(O_{st}\\)\u8868\u793a\\( s\\times t\\)\u7684\u96f6\u77e9\u9635\uff0c\\( r \\)\u7b49\u4e8e\u77e9\u9635\\(A\\)\u7684\u79e9\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\\(n \\) \u9636\u77e9\u9635\\( A \\)\u53ef\u9006\uff0c\u5f53\u4e14\u4ec5\u5f53\\( A \\)\u7684\u79e9\u7b49\u4e8e\\( n \\)\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u5982\u4f55\u6c42\u89e3\u77e9\u9635\u7684\u9006\uff1a<\/mark>\u6709\u4e86\u8fd9\u4e9b\u7ed3\u8bba\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4e00\u4e2a\u53ef\u9006\u77e9\u9635\u8981\u6c42\u5176\u9006\u77e9\u9635\uff0c\u5219\u53ef\u4ee5\u6709\u4e9b\u529e\u6cd5\uff1a\u5373\u5bf9\u4e00\u4e2a\u77e9\u9635\u5b9e\u65bd\u4e00\u7cfb\u5217\u7684\u521d\u7b49\u53d8\u6362\uff0c\u5c06\u5176\u53d8\u4e3a\u5355\u4f4d\u77e9\u9635\u3002\u800c\u540c\u65f6\uff0c\u5728\u5f00\u59cb\u7684\u65f6\u5019\uff0c\u5c31\u5c06\u6240\u6709\u7684\u8fd9\u4e9b\u521d\u7b49\u53d8\u6362\u4f5c\u7528\u5728\u4e00\u4e2a\u5355\u4f4d\u77e9\u9635\u4e0a\u3002\u6700\u540e\uff0c\u5f53\u539f\u77e9\u9635\u53d8\u4e3a\u5355\u4f4d\u77e9\u9635\u7684\u65f6\u5019\uff0c\u540e\u9762\u7684\u5355\u4f4d\u77e9\u9635\u5c31\u53d8\u6210\u539f\u77e9\u9635\u7684\u9006\u4e86\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark> \u4e24\u4e2a\u77e9\u9635\u4e58\u79ef\u7684\u79e9\uff0c\u4e0d\u5927\u4e8e\u4efb\u4f55\u4e00\u4e2a\u77e9\u9635\u7684\u79e9\u3002\u7279\u522b\u7684\uff0c\u5982\u679c\u6709\u4e00\u4e2a\u77e9\u9635\u662f\u53ef\u9006\u7684\uff0c\u5219\u4e58\u79ef\u7684\u79e9\u5219\u7b49\u4e8e\u53e6\u4e00\u4e2a\u77e9\u9635\u7684\u79e9\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark> \u4e00\u4e2a\u77e9\u9635\u7684\u884c\u7a7a\u95f4\u7684\u7ef4\u6570\u7b49\u4e8e\u5217\u7a7a\u95f4\u7684\u7ef4\u6570\uff0c\u7b49\u4e8e\u8fd9\u4e2a\u77e9\u9635\u7684\u79e9\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">5. \u5411\u91cf\u7a7a\u95f4<\/h3>\n\n\n\n<p>\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u524d\u9762\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\uff0c\u8fd8\u6709\u4e00\u4e9b\u95ee\u9898\u6ca1\u6709\u5f7b\u5e95\u56de\u7b54\uff08\u4f8b\u5982\uff0c\u89e3\u7a7a\u95f4\u7684\u63cf\u8ff0\uff09\uff0c\u5728\u56de\u7b54\u8fd9\u4e2a\u95ee\u9898\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u5148\u4e86\u89e3\u4e00\u4e0b\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u3002\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u7684\u4e25\u683c\u5b9a\u4e49\u662f\u6709\u4e9b\u67af\u71e5\u7684\uff0c\u8fd9\u91cc\u6682\u65f6\u628a\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u7684\u9650\u5236\u4e3a\u5927\u5bb6\u6240\u719f\u6089\u7684\u3001\u6700\u4e3a\u5178\u578b\u7684\u201c \\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\u201d\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">5.1 \u57fa<\/h4>\n\n\n\n<p>\u8981\u63cf\u8ff0\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\u7684\u5143\u7d20\uff0c\u5219\u9996\u5148\u9700\u8981\u4e00\u7ec4\u57fa\uff08\u5750\u6807\uff09\u3002\u5728\\(n\\)\u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff0c\u6700\u4e3a\u5e38\u89c1\u7684\u4e00\u7ec4\u57fa\uff0c\u5373\u4e3a\u591a\u4e2a\u201c\u5782\u76f4\u201d\uff08\u201c\u6b63\u4ea4\u201d\uff09\u7684\u5355\u4f4d\u5411\u91cf\uff0c\u5373\uff1a<\/p>\n\n\n<p>$$ \\begin{align}<br \/>\n\\alpha_1 &#038;= (1,0,\\dots) \\\\<br \/>\n&#038; \\vdots \\\\<br \/>\n\\alpha_i &#038;= (\\dots,0,1,0,\\dots) \\\\<br \/>\n&#038; \\vdots \\\\<br \/>\n\\alpha_n &#038;= (0,\\dots,1)<br \/>\n\\end{align}<br \/>\n$$<\/p>\n\n\n\n<p>\u5728\u4e8c\u7ef4\u5e73\u9762\u7a7a\u95f4\u4e2d\uff0c\u5219\u4e3a\uff1a\\( \\alpha_1 = (1,0) \\quad \\alpha_2 = (0,1)\\)\uff1b\u4e09\u7ef4\u7a7a\u95f4\u5219\u4e3a\uff1a\\( \\alpha_1 = (1,0,0) \\quad \\alpha_2 = (0,1,0) \\quad \\alpha_3 = (0,0,1)\\)\u3002<\/p>\n\n\n\n<p>\u65e2\u7136\u6709\u201c\u6b63\u4ea4\u201d\u57fa\uff0c\u90a3\u4e48\u5f53\u7136\u6709\u4e0d\u90a3\u4e48\u201c\u6b63\u4ea4\u201d\u7684\u57fa\uff0c\u800c\u6b64\u7c7b\u201c\u57fa\u201d\u5219\u662f\u66f4\u4e3a\u666e\u904d\u7684\u3002\u4e8b\u5b9e\u4e0a\uff0c\u66f4\u4e3a\u666e\u904d\u7684\uff0c\u4efb\u4f55 \\( n \\) \u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\u90fd\u53ef\u4ee5\u4f5c\u4e3a\u5411\u91cf\u7a7a\u95f4\u7684\u57fa\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">5.2 \u7ebf\u6027\u76f8\u5173\u4e0e\u7ebf\u6027\u65e0\u5173<\/h4>\n\n\n\n<p>\u8003\u8651\u4e00\u7ec4\u5411\u91cf\\( \\alpha_1,\\dots , \\alpha_n \\)\uff0c\u5982\u679c\u5f53\u4e14\u4ec5\u5f53\u6240\u6709\\( a_i = 0 \\quad i=1,\\dots,n\\)\u65f6\u5982\u4e0b\u7684\u7b49\u5f0f\u624d\u6210\u7acb\uff1a<\/p>\n\n\n<p>$$ a_1\\alpha_1 + a_2\\alpha_2 + \\dots + a_n\\alpha_n = 0 $$<\/p>\n\n\n\n<p>\u90a3\u4e48\uff0c\u5c31\u8bf4 \u8fd9\u7ec4\u5411\u91cf \\( \\alpha_1,\\dots , \\alpha_n \\) \u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002\u53cd\u4e4b\uff0c\u5219\u79f0\u8fd9\u7ec4\u5411\u91cf\u662f\u7ebf\u6027\u76f8\u5173\u7684\u3002<\/p>\n\n\n\n<p>\u6216\u8005\u8fd9\u4e48\u8bf4\uff0c\u5bf9\u4e8e\u4e00\u7ec4\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\\( \\alpha_1,\\dots , \\alpha_n \\)\uff1a\u4efb\u4f55\u4e00\u4e2a\u5411\u91cf\u90fd\u4e0d\u80fd\u7528\u5269\u4f59\u7684\u5411\u91cf\u505a\u201c\u7ebf\u6027\u8868\u793a\u201d\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">5.3 \u4e00\u4e9b\u91cd\u8981\u7684\u7ed3\u8bba<\/h4>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\u8bbe\\( \\{ \\alpha_1,\\dots , \\alpha_n \\}\\)\u662f\u5411\u91cf\u7a7a\u95f4\\( V \\)\u7684\u4e00\u7ec4\u57fa\uff0c\u90a3\u4e48\\( V \\)\u7a7a\u95f4\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5411\u91cf\u90fd\u53ef\u4ee5<strong>\u552f\u4e00<\/strong>\u7684\u8868\u793a\u4e3a\u8fd9\u7ec4\u57fa\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u8fd9\u4e2a\u7ebf\u6027\u7ec4\u5408\u7684\u7cfb\u6570\uff0c\u5c31\u53eb\u201c\u5750\u6807\u201d\uff08\u6ce8\uff1a\u76f8\u5bf9\u4e8e\u8fd9\u7ec4\u57fa\uff09\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\\( W_1 \\)\u3001\\( W_2 \\)\u662f\\( V \\)\u7684\u6709\u9650\u5b50\u7a7a\u95f4\uff0c\u90a3\u4e48\u6709\uff1a<\/p>\n\n\n<p>$$  dim(W_1+W_2) = dim(W_1) + dim(W_2) &#8211;  dim(W_1\\cap W_2)  $$<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\\( n \\)\u7ef4\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u4efb\u610f\\( n \\)\u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\u90fd\u53ef\u4ee5\u53d6\u505a\u57fa\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">6. \u7ebf\u6027\u53d8\u6362\u4e0e\u77e9\u9635<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">6.1 \u6982\u8ff0<\/h4>\n\n\n\n<p>\u201c\u7ebf\u6027\u53d8\u6362\u201d\u662f\u6307\u5411\u91cf\u7a7a\u95f4\u4e2d\u4e00\u7c7b\u7279\u6b8a\u7684\u6620\u5c04 \\( \\sigma : R^n \\to R^m \\) \uff0c\u9700\u8981\u6ee1\u8db3\u6761\u4ef6\u662f\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\( \\sigma(\\xi + \\eta)  = \\sigma(\\xi) + \\sigma(\\eta) \\)<\/li>\n\n\n\n<li>\\( \\sigma(a\\xi)  = a\\sigma(\\xi) \\)<\/li>\n<\/ul>\n\n\n\n<p>\u201c\u7ebf\u6027\u53d8\u6362\u201d \u63cf\u8ff0\u4e86\u5411\u91cf\u7a7a\u95f4\u4e4b\u95f4\u7684\u6620\u5c04\u3002\u540e\u7eed\u6240\u6709\u7684\u5185\u5bb9\u5927\u6982\u90fd\u662f\u56f4\u7ed5\u6b64\u800c\u5c55\u5f00\uff0c\u540e\u7eed\u6240\u6709\u7684\u5185\u5bb9\u90fd\u4f1a\u5c1d\u8bd5\u901a\u8fc7\u5404\u79cd\u65b9\u5f0f\u5c06 \u201c\u7ebf\u6027\u53d8\u6362\u201d \u7684\u7279\u6027\u7814\u7a76\u6e05\u695a\u3002\u8fd9\u91cc\u5199\u51fa\u90e8\u5206\u7ed3\u8bba\uff0c\u540e\u9762\u518d\u6162\u6162\u5c55\u5f00\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7ebf\u6027\u53d8\u6362\u4e4b\u4e0b\uff0c\u539f\u70b9\u4fdd\u6301\u4e0d\u53d8\u3002\u5373 \\( \\sigma( \\vec{0} ) = \\vec{0}  \\)<\/li>\n\n\n\n<li>\u51e0\u4f55\u610f\u4e49\u4e0b\uff0c\u901a\u5e38\uff0c\u7ebf\u6027\u53d8\u6362\u5305\u62ec\u4e86\uff1a\u65cb\u8f6c\u3001\u955c\u50cf\u3001\u62c9\u4f38\/\u538b\u7f29\uff08\u7279\u522b\u7684\uff0c\u6709\u65f6\u5019\u4f1a\u538b\u7f29\u5230\u96f6\uff09\u3001\u526a\u5207<\/li>\n<\/ul>\n\n\n\n<p>\u4e3a\u4e86\u7814\u7a76\u6e05\u695a\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u4e0a\u8ff0\u7684\u7279\u70b9\uff0c\u901a\u5e38\u9700\u8981\u9009\u53d6\u4e00\u7ec4\u201c\u57fa\u201d\uff0c\u7136\u540e\u4f7f\u7528\u8fd9\u7ec4\u201c\u57fa\u201d\u7684\u201c\u5750\u6807\u201d\u6765\u63cf\u8ff0\u7a7a\u95f4\u4e2d\u7684\u70b9\uff0c\u8fdb\u800c\u518d\u63cf\u8ff0\u5bf9\u5e94\u7684\u7ebf\u6027\u53d8\u6362\u3002\u6700\u4e3a\u5e38\u89c1\u7684\u57fa\u4e3a\u201c\u6b63\u4ea4\u5355\u4f4d\u57fa\u201d\u3002<\/p>\n\n\n\n<p>\u4ece\u65b9\u6cd5\u4e0a\u6765\u770b\uff0c\u7814\u7a76\u6e05\u695a\u201c\u7ebf\u6027\u53d8\u6362\u201d\u6700\u4e3a\u5173\u952e\u7684\u662f\u7814\u7a76\u6e05\u695a\u5bf9\u5e94\u7684\u201c\u53d8\u6362\u77e9\u9635\u201d\u3002\u6240\u4ee5\uff0c\u201c\u7ebf\u6027\u4ee3\u6570\u201d\u7684\u6838\u5fc3\u540e\u9762\u5c31\u53d8\u6210\u4e86\u5bf9\u77e9\u9635\u7279\u6027\u7684\u7814\u7a76\uff0c\u4f46\u662f\uff0c\u4e5f\u4e0d\u8981\u5fd8\u8bb0\u4e86\u521d\u8877\uff0c\u5426\u5219\u5f88\u5feb\u5c31\u8ff7\u5931\u4e86\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">6.2 \u7ebf\u6027\u53d8\u6362<\/h4>\n\n\n\n<p>\u6211\u5728\u5927\u5b66\u671f\u95f4\u5bf9\u4e8e\u7ebf\u6027\u53d8\u6362\u3001\u77e9\u9635\u6709\u4ec0\u4e48\u4f5c\u7528\uff0c\u662f\u5b8c\u5168\u6ca1\u6709\u6982\u5ff5\u7684\u3002\u6240\u4ee5\u5bf9\u4e8e\u4ed6\u4eec\u7684\u7279\u6027\u7814\u7a76\u4e5f\u6ca1\u6709\u638c\u63e1\u7684\u5f88\u6df1\uff0c\u57fa\u672c\u4e0a\u662f\u505c\u7559\u5728\u80fd\u591f\u628a\u4e00\u4e9b\u8054\u7cfb\u9898\u505a\u5bf9\u8fd9\u4e2a\u5c42\u9762\u3002\u800c\u73b0\u5728\uff0c\u6ce8\u610f\u7ebf\u6027\u53d8\u6362\u7684\u5e7f\u6cdb\u5e94\u7528\u4e4b\u540e\uff0c\u5c1d\u8bd5\u53bb\u7406\u89e3\u53bb\u672c\u8d28\u4e4b\u540e\uff0c\u5c31\u4f1a\u5bfb\u6839\u95ee\u5e95\u7684\u53bb\u7406\u89e3\u6e05\u695a\u4ec0\u4e48\u662f\u7ebf\u6027\u53d8\u6362\u3001\u4ec0\u4e48\u662f\u77e9\u9635\u3002\u8fd9\u91cc\u518d\u6b21\u8bf4\u8bf4\u6211\u7684\u7406\u89e3\u3002<\/p>\n\n\n\n<p>\u5728\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u6700\u4e3a\u5e38\u89c1\u7684\u662f \\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff0c\u4f1a\u6709\u5f88\u591a\u7684\u5411\u91cf\uff0c\u4f8b\u5982\u6bcf\u4e2a Embedding \u5c31\u53ef\u4ee5\u7406\u89e3\u662f\u5728\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4\u4e2d\uff0c<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u201c\u7ebf\u6027\u53d8\u6362\u201d\u8868\u8ff0<\/mark>\u4e86\u662f\u7a7a\u95f4\u4e2d\u7684\u4e00\u7c7b\u6620\u5c04\uff0c\u8be5\u6620\u5c04\u6ee1\u8db3\u4e0a\u8ff0\u201c\u5c0f\u7ed32.1\u201d\u4e2d\u7684\u4e24\u4e2a\u8981\u6c42\uff0c\u5373\u539f\u70b9\u4f9d\u65e7\u5230\u539f\u70b9\u3001\u6620\u5c04\u4fdd\u6301\u6240\u8c13\u7684\u201c\u7ebf\u6027\u201d\uff08\u4f8b\u5982\uff0c\u5411\u91cf\u548c\u7684\u6620\u5c04\u7b49\u4e8e\u6620\u5c04\u7684\u548c\u7b49\uff09\u3002<\/p>\n\n\n\n<p>\u5728\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5c31\u662f\u4e00\u4e2a\u7b26\u5408\u4e00\u5b9a\u6761\u4ef6\u7684\u6620\u5c04\u3002\u4e0e\u5411\u91cf\u7a7a\u95f4\u7684\u57fa\u7684\u9009\u53d6\u662f\u6ca1\u6709\u5173\u7cfb\u7684\u3002\u81ea\u6b64\uff0c\u4e0e\u77e9\u9635\u4e5f\u662f\u6ca1\u6709\u5173\u7cfb\u7684\u3002\u6240\u4ee5\uff0c\u7ebf\u6027\u53d8\u6362\u672c\u8eab\u662f\u66f4\u4e3a\u5e95\u5c42\u3001\u66f4\u4e3a\u57fa\u7840\u7684\u6982\u5ff5\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">6.3 \u6b27\u6c0f\u7a7a\u95f4\u7684\u7ebf\u6027\u53d8\u6362\u4e0e\u77e9\u9635<\/h4>\n\n\n\n<p>\u73b0\u5728\u6211\u4eec\u628a\u95ee\u9898\u9650\u5b9a\u5728 \\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u3002\u90a3\u4e48\uff0c\u8fd9\u65f6\u5019\uff0c\u6211\u4eec\u5982\u4f55\u63cf\u8ff0\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5462\uff1f\u662f\u7684\uff0c\u5c31\u662f\u201c\u57fa\u201d\u4e0e\u201c\u77e9\u9635\u201d\u3002<\/p>\n\n\n\n<p>\u901a\u5e38\uff0c\\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff0c\u6211\u4eec\u4f1a\u5148\u9009\u53d6\u4e00\u7ec4\u57fa\uff0c\u7136\u540e\u4f7f\u7528\u4e00\u4e2a\u77e9\u9635\u53bb\u63cf\u8ff0\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\u3002\u5e76\u4e14\u975e\u5e38\u5e78\u8fd0\u7684\uff0c\u4e00\u65e6\u8fd9\u7ec4\u57fa \u9009\u5b9a\u4e86\uff0c\u8fd9\u4e2a\u77e9\u9635\u662f\u552f\u4e00\u7684\u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\u5728\\(n \\) \u7ef4\u6b27\u6c0f\u7a7a\u95f4\uff08\u8fd9\u4e2a\u6761\u4ef6\u4f3c\u4e4e\u53ef\u4ee5\u53bb\u6389\uff09\u4e2d\uff0c\u5bf9\u4e8e\u7ebf\u6027\u53d8\u6362 \\(\\sigma \\)\uff0c\u5982\u679c\u9009\u5b9a\u4e00\u7ec4\u201c\u57fa\u201d\uff0c\u90a3\u4e48\u5c31\u5b58\u5728\u552f\u4e00\u7684\u201c\u77e9\u9635\u201d\u63cf\u8ff0\u8be5\u7ebf\u6027\u53d8\u6362\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7684\u7ed3\u8bba\uff0c\u662f\u6bd4\u8f83\u660e\u663e\u7684\u3002\u6211\u4eec\u8003\u8651\u5bf9\u4e8e\u7ebf\u6027\u53d8\u6362\u4e2d\u7684\u4e0a\u8ff0\u9009\u5b9a\u7684\u57fa\u5411\u91cf \\(\\alpha_i,\\quad \\text{where } i = 1,\\ldots,n \\)\uff0c\u7ebf\u6027\u53d8\u6362\u5c06\u5176\u6620\u5c04\u5230 \\(\\beta_i,\\quad \\text{where } i = 1,\\ldots,n \\)\uff0c\u90a3\u4e48\u6839\u636e\u201c\u57fa\u201d\u7684\u57fa\u672c\u6027\u8d28\uff0c\u5bf9\u4e8e\u8fd9\u91cc\u7684\u4efb\u4f55 \\(\\beta_i \\)\u90fd\u53ef\u4ee5\u8868\u793a\u6210\\(\\alpha_i \\)\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u6240\u6709\u7684\u8fd9\u4e9b\u7cfb\u6570\u6784\u6210\u7684\u77e9\u9635\uff0c\u5c31\u662f\u4e0a\u8ff0\u63cf\u8ff0\u7684\u552f\u4e00\u7684\u201c\u77e9\u9635\u201d\u3002\u5177\u4f53\u7684\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\n\\beta_1 &#038;= a_{11}\\alpha_1 + a_{21}\\alpha_2 + \\cdots + a_{n1}\\alpha_n \\\\[0.5em]<br \/>\n\\beta_2 &#038;= a_{12}\\alpha_1 + a_{22}\\alpha_2 + \\cdots + a_{n2}\\alpha_n \\\\[0.5em]<br \/>\n&#038;\\ \\vdots \\\\[0.5em]<br \/>\n\\beta_n &#038;= a_{1n}\\alpha_1 + a_{2n}\\alpha_2 + \\cdots + a_{nn}\\alpha_n<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u5373\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\n\\begin{bmatrix}<br \/>\n\\beta_1 &#038; \\beta_2 &#038; \\cdots &#038; \\beta_n<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\begin{bmatrix}<br \/>\n\\alpha_1 &#038; \\alpha_2 &#038; \\cdots &#038; \\alpha_n<br \/>\n\\end{bmatrix}<br \/>\n\\begin{bmatrix}<br \/>\na_{11} &#038; a_{12} &#038; \\cdots &#038; a_{1n} \\\\<br \/>\na_{21} &#038; a_{22} &#038; \\cdots &#038; a_{2n} \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\na_{n1} &#038; a_{n2} &#038; \\cdots &#038; a_{nn}<br \/>\n\\end{bmatrix}<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u5373\u5728\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\\(\\sigma \\) \u5728\u9009\u5b9a\u57fa \\(\\alpha_i,\\quad \\text{where } i = 1,\\ldots,n \\) \u5bf9\u5e94\u7684\u77e9\u9635\u4e3a\uff1a<\/p>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\nA =<br \/>\n\\begin{bmatrix}<br \/>\na_{11} &#038; a_{12} &#038; \\cdots &#038; a_{1n} \\\\<br \/>\na_{21} &#038; a_{22} &#038; \\cdots &#038; a_{2n} \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\<br \/>\na_{n1} &#038; a_{n2} &#038; \\cdots &#038; a_{nn}<br \/>\n\\end{bmatrix}<br \/>\n\\end{aligned}<br \/>\n$$<\/p>\n\n\n\n<p>\u4ece\u8fd9\u6bb5\u7b80\u6d01\u7684\u201c\u8bc1\u660e\u201d\uff08\u6216\u201c\u8bf4\u660e\u201d\uff09\u6765\u770b\uff0c\u6211\u4eec\u5f88\u81ea\u7136\u6709\u5982\u4e0b\u7ed3\u8bba\uff0c\u6839\u636e\u7a7a\u95f4\u4e2d\u7684\u201c\u57fa\u201d\u7684\u9009\u53d6\u4e0d\u540c\uff0c\u6211\u4eec\u4f1a\u5f97\u5230\u4e0d\u540c\u7684\u77e9\u9635\u3002\u56e0\u4e3a\u6211\u4eec\u53cd\u590d\u4f1a\u63d0\u5230\uff0c\u6211\u4eec\u7ecf\u5e38\u4f1a\u901a\u8fc7\u7814\u7a76\u77e9\u9635\u7684\u7279\u6027\u6765\u7814\u7a76\u7ebf\u6027\u53d8\u6362\u3002\u90a3\u4e48\uff0c\u540c\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5728\u4e0d\u540c\u7684\u201c\u57fa\u201d\u4e0b\u7684\u4e0d\u540c\u201c\u77e9\u9635\u201d\uff0c\u5f88\u81ea\u7136\u7684\u80fd\u591f\u60f3\u5230\uff0c\u8fd9\u4e9b\u201c\u77e9\u9635\u201d\u662f\u6709\u67d0\u4e9b\u5171\u6027\u7684\uff0c\u662f\u7684\uff0c\u6211\u4eec\u79f0\u8fd9\u4e9b\u77e9\u9635\u4e3a\u201c\u76f8\u4f3c\u77e9\u9635\u201d\uff0c\u76f8\u5173\u7279\u6027\uff0c\u6682\u4e0d\u5c55\u5f00\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">7. \u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">7.1 \u4e3a\u4ec0\u4e48<\/h4>\n\n\n\n<p>\u4e3a\u4ec0\u4e48\u6211\u4eec\u9700\u8981\u5173\u6ce8\u201c\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\u201d\u5462\uff1f\u4e3a\u4ec0\u4e48\u6211\u4eec\u8981\u53bb\u4e86\u89e3\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u5462\uff1f<\/p>\n\n\n\n<p>\u539f\u56e0\u662f\u201c\u7ebf\u6027\u53d8\u6362\u201d\u662f\u4e00\u4e2a\u6620\u5c04\uff0c\u662f\u975e\u5e38\u62bd\u8c61\u7684\u3002\u800c\u7279\u5f81\u503c \u3001\u7279\u5f81\u5411\u91cf\u3001SVD\u5206\u89e3\u53ef\u4ee5\u628a\u7ebf\u6027\u53d8\u6362\u6700\u4e3a\u5173\u952e\u7684\u7279\u6027\uff0c\u4ee5\u975e\u5e38\u201c\u76f4\u89c2\u201d\u7684\u5f62\u5f0f\u8868\u8fbe\u51fa\u6765\u3002\u5f53\u7136\uff0c\u8fd9\u91cc\u7684\u201c\u76f4\u89c2\u201d\u5e76\u4e0d\u662f\u7b80\u5355\u610f\u4e49\u4e0a\u80fd\u591f\u4e00\u773c\u5c31\u770b\u51fa\u4ec0\u4e48\u6765\uff0c\u4e8b\u5b9e\u4e0a\uff0c\u201c\u7ebf\u6027\u53d8\u6362\u201d\u672c\u8eab\u5c31\u6709\u5f88\u5f3a\u7684\u62bd\u8c61\u6027\uff0c\u8fd9\u91cc\u7684\u201c\u76f4\u89c2\u201d\u53ea\u662f\u76f8\u5bf9\u7684\uff0c\u662f\u5426\u76f4\u89c2\uff0c\u5b8c\u5168\u4f9d\u8d56\u4e8e\u5404\u4f4d\u770b\u5ba2\u81ea\u5df1\u7684\u201c\u609f\u6027\u201d\u4e86 \u3002<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7279\u5f81\u5411\u91cf\u7684\u57fa\u672c\u5b9a\u4e49\uff1a<\/mark>\u5982\u679c\u6709 \\( \\sigma(\\xi) = \\lambda \\xi \\) \uff0c\u90a3\u4e48\u8fd9\u91cc\u7684  \\( \\xi \\) \u5c31\u662f\u7279\u5f81\u5411\u91cf\uff0c\u5bf9\u5e94\u7684 \\(\\lambda \\) \u5c31\u662f\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u3002<\/p>\n\n\n\n<p>\u8981\u60f3\u771f\u6b63\u8bf4\u6e05\u695a\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c\u662f\u9700\u8981\u975e\u5e38\u591a\u7bc7\u5e45\u7684\uff0c\u800c\u4e14\u5173\u4e8e\u5bf9\u7279\u5f81\u5411\u91cf\u7684\u7406\u89e3\u5bf9\u4e8e\u7406\u89e3\u7ebf\u6027\u53d8\u5316\u4e5f\u662f\u975e\u5e38\u5173\u952e\u7684\uff0c\u6240\u4ee5\uff0c\u5efa\u8bae\u82b1\u4e9b\u65f6\u95f4\u8f83\u4e3a\u7cfb\u7edf\u7684\u505a\u4e00\u4e9b\u7406\u89e3\u3002\u5982\u679c\u4f60\u5df2\u7ecf\u5efa\u7acb\u7684\u57fa\u7840\u6982\u5ff5\uff0c\u8fd9\u91cc\u7684\u4e00\u7bc7\u6587\u7ae0\u53ef\u80fd\u662f\u5e2e\u52a9\u4f60\u589e\u5f3a\u4e00\u4e9b\u7406\u89e3\uff1a\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">7.2 \u5173\u4e8e\u5bf9\u7279\u5f81\u5411\u91cf\u7684\u7406\u89e3<\/h4>\n\n\n\n<p>\u5b8c\u6574\u7684\u8ba8\u8bba\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u503c\u662f\u590d\u6742\u7684\uff0c\u8fd9\u91cc\u5c06\u5176\u9650\u5b9a\u5728\u4e00\u4e9b\u8f83\u4e3a\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4f5c\u4e3a\u4e00\u4e2a\u5165\u95e8\u3002\u6211\u4eec\u8fd9\u91cc\u8003\u8651\u6700\u4e3a\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u5373\u5bf9\u4e8e\u4e00\u4e2a \\(n \\times n \\)\u7684\u77e9\u9635\uff0c\u5176\u79e9\u4e3a \\(n \\)\uff0c\u5e76\u4e14\u5728\u8ba1\u7b97\u7279\u5f81\u503c\u65f6\uff0c\u6709 \\(n \\) \u662f\u4e0d\u91cd\u590d\u7684\u5b9e\u6570\u89e3\uff0c\u5373\u6ca1\u6709\u4efb\u4f55\u6839\u5f0f\u91cd\u6839\u3002\u5982\u679c\uff0c\u6070\u597d \\( n = 2 \\)\u8fd9\u5927\u6982\u662f\u6700\u4e3a\u7b80\u5355\u7684\u60c5\u51b5\u4e86\uff0c\u4e0d\u8fc7\u7406\u89e3\u8fd9\u79cd\u60c5\u51b5\uff0c\u518d\u8fdb\u4e00\u6b65\u62d3\u5c55\uff0c\u5219\u5b66\u4e60\u66f2\u7ebf\u4f1a\u5e73\u6ed1\u5f88\u591a\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u6765\u770b\u4e00\u4e2a\u5b9e\u4f8b\uff0c\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u5728\u6807\u51c6\u57fa\u4e0b\uff0c\u6211\u4eec\u6709\u5982\u4e0b\u7684\u7ebf\u6027\u53d8\u6362\u77e9\u9635\uff1a<\/p>\n\n\n<p>$$<br \/>\nW = \\begin{bmatrix}<br \/>\n2 &#038; 1 \\\\<br \/>\n1 &#038; 2<br \/>\n\\end{bmatrix}<br \/>\n$$<\/p>\n\n\n\n<p>\u6839\u636e\u4e0a\u8ff0\u7279\u5f81\u5411\u91cf\u7279\u5f81\u503c\u7684\u5b9a\u4e49\u8fdb\u884c\u6c42\u89e3\uff0c\u6211\u4eec\u53ef\u4ee5\u6709\u5982\u4e0b\u7684\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\( \\lambda_1 = 3 \\) \u7279\u5f81\u5411\u91cf \\( (1,1) \\)<\/li>\n\n\n\n<li>\\( \\lambda_2 = 1 \\) \u7279\u6027\u5411\u91cf \\( (-1,1) \\)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image alignright size-full is-resized\" style=\"margin-right:var(--wp--preset--spacing--small)\"><img loading=\"lazy\" decoding=\"async\" width=\"694\" height=\"682\" src=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-18.png\" alt=\"\" class=\"wp-image-22851\" style=\"width:316px;height:auto\" srcset=\"https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-18.png 694w, https:\/\/www.orczhou.com\/wp-content\/uploads\/2026\/01\/image-18-300x295.png 300w\" sizes=\"auto, (max-width: 694px) 100vw, 694px\" \/><\/figure>\n\n\n\n<p>\u4ece\u7279\u5f81\u5411\u91cf\u89d2\u5ea6\u7406\u89e3\u7ebf\u6027\u53d8\u6362\uff1a\u90a3\u4e48\u4e0a\u8ff0\u7684\u77e9\u9635A\u5bf9\u5e94\u7684\u7ebf\u6027\u53d8\u6362 \\(\\sigma \\) \u6709\u5982\u4e0b\u7279\u6027\uff0c\u5728\u8fd9\u4e2a\u4e8c\u7ef4\u7a7a\u95f4\u4efb\u4f55\u5411\u91cf \\(\\beta \\)\uff0c\u90fd\u53ef\u4ee5\u5206\u89e3\uff08\u6295\u5f71\uff09\u4e3a\u4e0a\u8ff0\u4e24\u4e2a\u7279\u5f81\u5411\u91cf\u65b9\u5411\u7684\u5411\u91cf\uff1a \\(\\beta_1 \\,, \\beta_2 \\)\uff0c\u4e14\u6709\uff1a\\(\\beta = \\beta_1 + \\beta_2 \\)\u3002\u90a3\u4e48\uff0c\u5219\u6709\uff1a\\(\\sigma(\\beta) = \\lambda_1  \\beta_1 + \\lambda_2  \\beta_2  \\)\u3002\u5373\uff0c\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\u53ef\u4ee5\u8fd9\u6837\u63cf\u8ff0\uff1a\u5148\u5c06\u4efb\u4f55\u5411\u91cf\u6cbf\u7740\u7279\u5f81\u5411\u91cf\u65b9\u5411\u5206\u89e3\uff0c\u7136\u540e\u518d\u6309\u7167\u7279\u5f81\u503c\u7684\u5927\u5c0f\u8fdb\u884c\u62c9\u4f38\u6216\u538b\u7f29\uff0c\u7136\u540e\u518d\u628a\u5411\u91cf\u5408\u5e76\u8d77\u6765\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7684\u89e3\u91ca\uff0c\u53ef\u4ee5\u5bf9\u7167\u7740\u53f3\u56fe\u53bb\u7406\u89e3\u3002\u7279\u5f81\u5411\u91cf\u5206\u522b\u4e3a \\((1,1) \\) \u548c \\((-1,1) \\) \uff0c\u5373\u56fe\u4e2d\u6d45\u7eff\u8272\u3001\u6d45\u84dd\u8272\u65b9\u5411\u3002\u8be5\u77e9\u9635\u4f5c\u7528\u5728\u5411\u91cf \\((0,1) \\) \u4e0a\uff0c\u5373\u56fe\u4e2d\u7684\u7ea2\u8272\u5411\u91cf\u3002\u5148\u5c06\u7ea2\u8272\u5411\u91cf\u6cbf\u7740\u6d45\u7eff\u8272\u3001\u6d45\u84dd\u8272\u65b9\u5411\u5206\u89e3\uff0c\u7136\u540e\u6309\u7167\u7279\u5f81\u503c\u8fdb\u884c\u62c9\u4f38\uff0c\u5373\u56fe\u4e2d\u7684\u7eff\u8272\u3001\u84dd\u8272\u5411\u91cf\uff0c\u6700\u540e\u5408\u5e76\u4e3a\u56fe\u4e2d\u7684\u7d2b\u8272\u6700\u7ec8\u5411\u91cf\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7684\u573a\u666f\u662f\u7ebf\u6027\u53d8\u6362\u4e2d\uff0c\u6700\u4e3a\u7b80\u5355\u7684\u4e00\u7c7b\u3002\u800c\u5b9e\u9645\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u5219\u66f4\u4e3a\u590d\u6742\uff0c\u53ef\u80fd\u8fd8\u6d89\u53ca\u5230\u5bf9\u4e8e\u5411\u91cf\u7684\u65cb\u8f6c\u3001\u955c\u50cf\u3001\u526a\u5207\u7b49\u53d8\u6362\u3002\u5173\u4e8e\u66f4\u591a\u573a\u666f\u53ef\u4ee5\u81ea\u5df1\u63a2\u7d22\uff0c\u6216\u8005\u9605\u8bfb\u76f8\u5173\u4e66\u7c4d\uff0c\u4e5f\u53ef\u4ee5\u770b\u770b\u8fd9\u7bc7\u6587\u7ae0\u4e2d\u7684\u66f4\u591a\u76f4\u89c2\u7684\u4f8b\u5b50\uff1a<a href=\"https:\/\/www.orczhou.com\/?p=19875&amp;preview=true\">\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf<\/a>\u3002<\/p>\n\n\n\n<p>\u201c\u7279\u5f81\u5411\u91cf\u201d\u53ef\u4ee5\u5f88\u597d\u7684\u5e2e\u52a9\u7406\u89e3\u201c\u65b9\u9635\u201d\u53d8\u6362\uff0c\u8fd8\u6709\u4e00\u7c7b\u53d8\u6362\u65f6\u975e\u65b9\u9635\u7684\u60c5\u51b5\uff0c\u901a\u5e38\u8fd9\u65f6\u5019\u53ef\u4ee5\u501f\u52a9\u4e8e\u5947\u5f02\u503c\u5206\u89e3\u7684\u65b9\u5f0f\u53bb\u7406\u89e3\uff0c\u5173\u4e8e\u5947\u5f02\u503c\u5206\u89e3\u53ef\u4ee5\u53c2\u8003\uff1a<a href=\"https:\/\/www.orczhou.com\/index.php\/2025\/09\/svd-01\/\">\u5947\u5f02\u503c\u5206\u89e3\u2013\u6df1\u5ea6\u5b66\u4e60\u7684\u6570\u5b66\u57fa\u7840<\/a>\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">8. \u4e00\u4e9b\u8865\u5145\u8bf4\u660e<\/h3>\n\n\n\n<p>\u521d\u7b49\u7684\u7ebf\u6027\u4ee3\u6570\u6838\u5fc3\u90e8\u5206\u5927\u6982\u662f\u8fd9\u4e9b\u5185\u5bb9\uff0c\u51fa\u4e8e\u5b8c\u6574\u6027\u7684\u8003\u8651\uff0c\u53ef\u4ee5\u518d\u8fdb\u4e00\u6b65\u4e86\u89e3\u201cJordan \u5757\u201d\u76f8\u5173\u7684\u5185\u5bb9\uff0c\u4ece\u800c\u628a\u76f8\u5173\u7406\u8bba\u8865\u5145\u5b8c\u6574\uff0c\u8fd9\u91cc\u4e0d\u518d\u8be6\u8ff0\u3002<\/p>\n\n\n\n<p>\u5982\u679c\u518d\u56de\u5230\u6700\u521d\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u95ee\u9898\uff0c\u6211\u4eec\u8fd9\u91cc\u5c31\u53ef\u4ee5\u56de\u7b54\u6700\u540e\u4e00\u4e2a\u95ee\u9898\uff1a\u5bf9\u4e8e\u4e00\u4e2a\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u5982\u679c\u6709\u89e3\uff0c\u90a3\u4e48\u6240\u6709\u7684\u89e3\u7a7a\u95f4\u662f\u600e\u6837\u7684\uff1f<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-primary-color\">\u7ed3\u8bba\uff1a<\/mark>\u5982\u679c\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635\u7684\u79e9\u4e3a\\( r \\)\uff0c\u90a3\u4e48\u89e3\u7a7a\u95f4\u7684\u7ef4\u5ea6\u4e3a\\( n-r \\)\u3002\u89e3\u7a7a\u95f4\u7684\u201c\u57fa\u201d\u5219\u53ef\u4ee5\u901a\u8fc7\u521d\u7b49\u53d8\u6362\u6c42\u5f97\u3002\u8fd9\u91cc\u4e0d\u518d\u8be6\u8ff0\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">9. \u518d\u770b\u770b\u524d\u9762\u7684\u95ee\u9898<\/h3>\n\n\n<p>$$<br \/>\n\\begin{aligned}<br \/>\n\\text{Attention Score}_i = XW_i^{QK} X^T \\quad \\text{where} \\,,W_i^{QK} = W_i^Q (W_i^K)^T<br \/>\n\\end{aligned}\\tag{1}<br \/>\n$$<\/p>\n\n\n<p>$$<br \/>\n\\text{Attention Score}_i = Q_iK_i^T = XW_i^Q(XW_i^K)^T  \\tag{2}<br \/>\n$$<\/p>\n\n\n\n<p>\u5728\u4e0a\u9762\u7684\u8ba1\u7b97(1) \u548c \u8ba1\u7b97(2)\uff0c\u90a3\u4e48\u5728\u6a21\u578b\u4e2d\u8bad\u7ec3 \\(W_i^{QK} \\) \u548c \u5728\u6a21\u578b\u4e2d\u8bad\u7ec3\u5355\u72ec\u8bad\u7ec3 \\(W_i^Q  \\,,W_i^K  \\) \u662f\u5426\u662f\u7b49\u4ef7\u7684\uff1f<\/p>\n\n\n\n<p>\u7b54\u6848\u662f\u5426\u5b9a\u7684\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc\u4ee5 GTP2 \u6a21\u578b\u4e3a\u4f8b\uff0c\u539f\u56e0\u5728\u4e8e\u5982\u679c\u5355\u72ec\u8bad\u7ec3 \\(W_i^{QK} \\) \uff0c\u90a3\u4e48\u8fd9\u4e2a\u77e9\u9635\u7684\u79e9\uff0c\u5219\u5f88\u53ef\u80fd\u662f 768 \uff1b\u800c\u5355\u72ec\u8bad\u7ec3 \\(W_i^Q  \\,,W_i^K  \\)\uff0c\u8fd9\u4e24\u4e2a\u77e9\u9635\u7684\u79e9\u5219\u4e00\u5b9a\u5c0f\u4e8e 64\uff0c\u8fd9\u4e24\u4e2a\u77e9\u9635\u7684\u4e58\u79ef\u7684\u79e9\u4e5f\u4e00\u5b9a\u662f 64 \uff08\u4e25\u683c\u6765\u8bf4\u662f\u5c0f\u4e8e\u7b49\u4e8e\uff09\u3002\u6240\u4ee5\u6700\u7ec8\u8bad\u7ec3\u83b7\u5f97\u7684\u6548\u679c\u4e00\u5b9a\u662f\u4e0d\u540c\u7684\u3002\u5f53\u7136\uff0c\u54ea\u4e2a\u66f4\u597d\uff0c\u8fd9\u5012\u4e0d\u4e00\u5b9a\uff0c\u4f46\u4ed6\u4eec\u5e76\u4e0d\u662f\u7b49\u4ef7\u7684\u3002<\/p>\n\n\n\n<p>\u4e00\u822c\u610f\u4e49\u6765\u8bf4\uff0c\u4f7f\u7528 \\(W_i^{QK} \\) \u53ef\u80fd\u6709\u7740\u66f4\u5f3a\u7684\u8868\u8fbe\u80fd\u529b\uff0c\u53ea\u662f\u610f\u4e49\u6ca1\u6709\u90a3\u4e48\u660e\u786e\uff0c\u5e76\u4e14\u8bad\u7ec3\u7684\u53c2\u6570\u8981\u66f4\u591a\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5982\u679c\u8981\u7406\u89e3\u5927\u6a21\u578b\u5185\u90e8\u662f\u5982\u4f55\u5de5\u4f5c\u7684\uff0c\u826f\u597d\u7684\u6570\u5b66\u57fa\u7840\u662f\u5fc5\u987b\u7684\uff0c\u800c\u7ebf\u6027\u4ee3\u6570\u53c8\u662f\u6240\u6709\u8fd9\u4e9b\u7684\u57fa\u7840\u3002\u4f8b\u5982\u6211\u4eec\u6765\u770b $latex \\text{Attention} $ 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