![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729171925460.png\")
Deep Learning Math<\/h1>\n\n
Deep Learning Math<\/h1>\n\u7ebf\u6027\u4ee3\u6570\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\u5360\u636e\u6838\u5fc3\u5730\u4f4d\u3002\u7ebf\u6027\u65b9\u7a0b\u7ec4\u548c\u589e\u5e7f\u77e9\u9635\u7528\u4e8e\u63cf\u8ff0\u548c\u89e3\u51b3\u7cfb\u7edf\u4e2d\u7684\u591a\u53d8\u91cf\u5173\u7cfb\u3002\u5411\u91cf\u64cd\u4f5c\uff0c\u5982\u5411\u91cf\u6570\u4e58\u3001\u52a0\u6cd5\u548c\u7ebf\u6027\u7ec4\u5408\uff0c\u6784\u6210\u4e86\u6df1\u5ea6\u5b66\u4e60\u4e2d\u6570\u636e\u548c\u6743\u91cd\u7684\u57fa\u672c\u8868\u793a\u3002\u5411\u91cf\u7a7a\u95f4\u548c\u7ebf\u6027\u76f8\u5173\/\u65e0\u5173\u7684\u6982\u5ff5\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u6570\u636e\u7684\u7ed3\u6784\u548c\u7ef4\u5ea6\u3002\u70b9\u79ef\u548c\u5916\u79ef\u5728\u8ba1\u7b97\u76f8\u4f3c\u5ea6\u548c\u751f\u6210\u65b0\u5411\u91cf\u65f6\u975e\u5e38\u91cd\u8981\uff0c\u800c\u6b63\u4ea4\u4e0e\u57fa\u7ec4\u5219\u7528\u4e8e\u7b80\u5316\u8ba1\u7b97\u548c\u51cf\u5c11\u7ef4\u5ea6\u3002<\/p>\n
\u77e9\u9635\u8fd0\u7b97\uff0c\u5982\u77e9\u9635\u4e58\u6cd5\u3001\u54c8\u8fbe\u739b\u79ef\u548c\u514b\u7f57\u5185\u514b\u79ef\uff0c\u662f\u6df1\u5ea6\u5b66\u4e60\u4e2d\u5f20\u91cf\u64cd\u4f5c\u7684\u57fa\u7840\u3002\u521d\u7b49\u77e9\u9635\u548c\u53ef\u9006\u77e9\u9635\u7684\u6982\u5ff5\u7528\u4e8e\u4f18\u5316\u548c\u6c42\u89e3\u6a21\u578b\u53c2\u6570\u3002\u884c\u5217\u5f0f\u548c\u77e9\u9635\u7684\u79e9\u5e2e\u52a9\u786e\u5b9a\u7cfb\u7edf\u7684\u89e3\u548c\u77e9\u9635\u7684\u5c5e\u6027\uff0c\u800c\u77e9\u9635\u7684\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\u5219\u7528\u4e8e\u7406\u89e3\u6570\u636e\u7684\u53d8\u6362\u548c\u964d\u7ef4\u3002\u901a\u8fc7\u8fd9\u4e9b\u7ebf\u6027\u4ee3\u6570\u5de5\u5177\uff0c\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u80fd\u591f\u9ad8\u6548\u5904\u7406\u548c\u8868\u793a\u5927\u91cf\u6570\u636e\uff0c\u5b9e\u73b0\u590d\u6742\u8ba1\u7b97\u548c\u6a21\u578b\u4f18\u5316\u3002<\/p>\n
![](\"https:\/\/img.icons8.com\/bubbles\/50\/000000\/checklist.png\")
Agenda<\/h3>\n\u7ebf\u6027\u4ee3\u6570\u7684\u96be\u70b9\u5728\u4e8e\u5176\u6d89\u53ca\u591a\u79cd\u8868\u793a\u7cfb\u7edf\uff0c\u5305\u62ec\u65b9\u7a0b\u7ec4\u8868\u793a\u3001\u5411\u91cf\u8868\u793a\u548c\u77e9\u9635\u8868\u793a\u7b49\u3002\u7406\u89e3\u548c\u638c\u63e1\u8fd9\u4e9b\u4e0d\u540c\u8868\u793a\u65b9\u6cd5\u4e4b\u95f4\u7684\u5173\u7cfb\u4e0e\u8f6c\u6362\u662f\u81f3\u5173\u91cd\u8981\u7684\uff0c\u56e0\u4e3a\u5b83\u4eec\u5728\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u65f6\u63d0\u4f9b\u4e86\u4e0d\u540c\u7684\u89c6\u89d2\u548c\u5de5\u5177\u3002\u4f8b\u5982\uff0c<\/p>\n
\u65b9\u7a0b\u7ec4\u8868\u793a\u901a\u5e38\u7528\u4e8e\u63cf\u8ff0\u548c\u6c42\u89e3\u7cfb\u7edf\u7684\u7ebf\u6027\u5173\u7cfb<\/p>\n<\/li>\n
\u5411\u91cf\u8868\u793a\u5219\u5f3a\u8c03\u51e0\u4f55\u548c\u4ee3\u6570\u7ed3\u6784<\/p>\n<\/li>\n
\u77e9\u9635\u8868\u793a\u63d0\u4f9b\u4e86\u5904\u7406\u548c\u8ba1\u7b97\u591a\u53d8\u91cf\u7ebf\u6027\u7cfb\u7edf\u7684\u5f3a\u5927\u6846\u67b6\u3002<\/p>\n<\/li>\n<\/ul>\n
\u5168\u9762\u7406\u89e3\u8fd9\u4e9b\u8868\u793a\u65b9\u6cd5\u53ca\u5176\u76f8\u4e92\u8f6c\u6362\uff0c\u4e0d\u4ec5\u6709\u52a9\u4e8e\u6df1\u5316\u5bf9\u7ebf\u6027\u4ee3\u6570\u7406\u8bba\u7684\u7406\u89e3\uff0c\u4e5f\u80fd\u591f\u63d0\u9ad8\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u7075\u6d3b\u8fd0\u7528\u8fd9\u4e9b\u5de5\u5177\u89e3\u51b3\u590d\u6742\u95ee\u9898\u7684\u80fd\u529b\u3002<\/p>\n
\u5148\u4ece\u7ebf\u6027\u65b9\u7a0b\u7ec4\u5f00\u59cb\u8bb2\u8d77\uff0c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u662f\u6d89\u53ca\u540c\u4e00\u7ec4\u53d8\u91cf\u7684\u65b9\u7a0b\u7684\u96c6\u5408\u3002\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u4e00\u822c\u5f62\u5f0f\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n
$$\\begin{cases}a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n=b_1\\\\a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n=b_2\\\\
\n\\vdots \\\\
\na_{m1}x_1+a_{m2}x_2+\\cdots+a_{mn}x_n=b_m\\end{cases}$$
\n\u5176\u4e2d $a_{i j}$ \u8868\u793a $i$ \u884c $j$ \u5217\u7684\u7cfb\u6570, $x_n$ \u8868\u793a\u53d8\u91cf\/\u672a\u77e5\u6570, $b_m$ \u8868\u793a\u5e38\u6570<\/p>\n
\u4e3e\u4e2a\u4f8b\u5b50, \u73b0\u5728\u6709\u65b9\u7a0b\u7ec4\u5982\u4e0b:<\/p>\n
$$
\n\\begin{cases}
\nx + 2y = 7 \\\\
\nx - y = 1
\n\\end{cases}
\n$$<\/p>\n
\u6bcf\u4e2a\u65b9\u7a0b\u90fd\u53ea\u6709\u4e24\u4e2a\u672a\u77e5\u6570, \u8fd9\u6837\u7684\u65b9\u7a0b\u5c31\u662f\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u6761\u76f4\u7ebf\u3002\u800c\u6c42\u542b\u6709\u4e24\u4e2a\u672a\u77e5\u6570\u7684\u4e24\u4e2a\u65b9\u7a0b\u7ec4\u6210\u7684\u65b9\u7a0b\u7ec4\u7684\u89e3, \u7b49\u4ef7\u4e8e\u6c42\u4e24\u6761\u76f4\u7ebf\u7684\u4ea4\u70b9\u3002\u5f88\u5bb9\u6613\u6c42\u51fa\u4ee5\u4e0a\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u4e3a $x=3, y=2$, \u56fe\u5f62\u7ed3\u679c\u5982\u4e0b\u6240\u793a\u3002<\/p>\n
import numpy as np\nimport matplotlib.pyplot as plt\n\n# \u5b9a\u4e49\u65b9\u7a0b\ndef equation1(x):\n return (7 - x) \/ 2\n\ndef equation2(x):\n return x - 1\n\n# \u5b9a\u4e49x\u8303\u56f4\nx = np.linspace(0, 5, 400)\n\n# \u8ba1\u7b97y\u503c\ny1 = equation1(x)\ny2 = equation2(x)\n\n# \u7ed8\u5236\u56fe\u5f62\nplt.figure(figsize=(8, 6))\nplt.plot(x, y1, label=r'$x + 2y = 7$')\nplt.plot(x, y2, label=r'$x - y = 1$')\n\n# \u6807\u8bb0\u4ea4\u70b9\nplt.scatter(3, 2, color='red') \nplt.text(3, 2, ' (3, 2)', fontsize=12, verticalalignment='bottom', horizontalalignment='right')\n\n# \u8bbe\u7f6e\u56fe\u5f62\u5c5e\u6027\nplt.axhline(0, color='black',linewidth=0.5)\nplt.axvline(0, color='black',linewidth=0.5)\nplt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)\nplt.legend()\nplt.title('Graphical Solution of Linear Equations')\nplt.xlabel('x')\nplt.ylabel('y')\nplt.xlim(0, 5)\nplt.ylim(0, 5)\nplt.show()<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_3_0.png\")
\n<\/p>\n
\u6b64\u65f6\uff0c\u65b9\u7a0b\u7ec4\u5b58\u5728\u4e00\u4e2a\u552f\u4e00\u89e3\u3002\u5f53\u7136\uff0c\u4e24\u6761\u76f4\u7ebf\u5e76\u4e0d\u4e00\u5b9a\u4ea4\u4e8e\u4e00\u70b9\uff0c\u5b83\u4eec\u53ef\u80fd\u5e73\u884c\uff0c\u4e5f\u53ef\u80fd\u91cd\u5408\uff0c\u91cd\u5408\u7684\u4e24\u6761\u76f4\u7ebf\u4e0a\u7684\u6bcf\u4e2a\u70b9\u90fd\u662f\u4ea4\u70b9\u3002\u8003\u8651\u4e0b\u9762\u4e24\u4e2a\u65b9\u7a0b\u7ec4\uff1a
\n$$
\n\\begin{cases} { l }
\n{ x - 2 y = - 1 } \\\\
\n{ - x + 2 y = 3 }
\n\\end{cases}
\n\\\\
\n\\begin{cases}{l}
\nx-2 y=-1 \\\\
\n-x+2 y=1
\n\\end{cases}
\n$$
\n\u5176\u4e2d\u7b2c\u4e00\u4e2a\u65b9\u7a0b\u7ec4\u4e2d\u7684\u4e24\u6761\u76f4\u7ebf\u5e73\u884c\uff0c\u6ca1\u6709\u4ea4\u70b9\uff0c\u5373\u65b9\u7a0b\u7ec4\u65e0\u89e3\uff1b\u7b2c\u4e8c\u4e2a\u65b9\u7a0b\u7ec4\u4e2d\u7684\u4e24\u6761\u76f4\u7ebf\u91cd\u5408\uff0c\u6709\u65e0\u6570\u4ea4\u70b9\uff0c\u5373\u65b9\u7a0b\u7ec4\u6709\u65e0\u7a77\u591a\u89e3\u3002<\/p>\n
import numpy as np\nimport matplotlib.pyplot as plt\n\n# \u5b9a\u4e49\u65b9\u7a0b\u7ec4\u7684\u65b9\u7a0b\ndef equation1a(x):\n return (x + 1) \/ 2\n\ndef equation1b(x):\n return (x - 3) \/ 2\n\ndef equation2a(x):\n return (x + 1) \/ 2\n\ndef equation2b(x):\n return (x + 1) \/ 2\n\n# \u5b9a\u4e49x\u8303\u56f4\nx = np.linspace(-2, 2, 400)\n\n# \u8ba1\u7b97y\u503c\ny1a = equation1a(x)\ny1b = equation1b(x)\ny2a = equation2a(x)\ny2b = equation2b(x)\n\n# \u7ed8\u5236\u56fe\u5f62\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))\n\n# \u7b2c\u4e00\u4e2a\u65b9\u7a0b\u7ec4\nax1.plot(x, y1a, label=r'$x - 2y = -1$')\nax1.plot(x, y1b, label=r'$-x + 2y = 3$')\nax1.set_title('No Solution')\nax1.axhline(0, color='black',linewidth=0.5)\nax1.axvline(0, color='black',linewidth=0.5)\nax1.grid(color = 'gray', linestyle = '--', linewidth = 0.5)\nax1.legend()\nax1.set_xlim(-2, 2)\nax1.set_ylim(-2, 2)\nax1.spines['left'].set_position('zero')\nax1.spines['bottom'].set_position('zero')\nax1.spines['right'].set_color('none')\nax1.spines['top'].set_color('none')\nax1.xaxis.set_ticks_position('bottom')\nax1.yaxis.set_ticks_position('left')\n\n# \u7b2c\u4e8c\u4e2a\u65b9\u7a0b\u7ec4\nax2.plot(x, y2a, label=r'$x - 2y = -1$')\nax2.plot(x, y2b, label=r'$-x + 2y = 1$')\nax2.set_title('Infinite Solutions')\nax2.axhline(0, color='black',linewidth=0.5)\nax2.axvline(0, color='black',linewidth=0.5)\nax2.grid(color = 'gray', linestyle = '--', linewidth = 0.5)\nax2.legend()\nax2.set_xlim(-2, 2)\nax2.set_ylim(-2, 2)\nax2.spines['left'].set_position('zero')\nax2.spines['bottom'].set_position('zero')\nax2.spines['right'].set_color('none')\nax2.spines['top'].set_color('none')\nax2.xaxis.set_ticks_position('bottom')\nax2.yaxis.set_ticks_position('left')\n\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_5_0.png\")
\n<\/p>\n
\u901a\u8fc7\u4e0a\u9762\u7684\u4f8b\u5b50\uff0c\u53ef\u4ee5\u603b\u7ed3\u4e00\u4e2a\u91cd\u8981\u7684\u7ed3\u8bba\uff1a\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u53ea\u6709\u4e09\u79cd\u60c5\u51b5\uff1a\u4e00\u4e2a\u89e3\u3001\u65e0\u7a77\u89e3\u548c\u65e0\u89e3\u3002<\/p>\n
\u73b0\u5728\u628a\u65b9\u7a0b\u6269\u5c55\u5230\u4e09\u4e2a\u672a\u77e5\u6570\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u8fd9\u6837\u6bcf\u4e2a\u65b9\u7a0b\u5c06\u786e\u5b9a\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5e73\u9762\u3002<\/p>\n
\u73b0\u5728\u60f3\u8c61\u4e00\u4e0b\u4e09\u4e2a\u8fd9\u6837\u7684\u5e73\u9762\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5206\u5e03\u4f1a\u6709\u51e0\u79cd\u60c5\u51b5\uff1f\u5176\u5b9e\u4e5f\u662f\u4e0a\u8ff0\u7684\u4e09\u79cd\u60c5\u51b5\uff1a<\/p>\n
\n- \u5f53\u4e09\u4e2a\u5e73\u9762\u76f8\u4e92\u5e73\u884c\u65f6\uff0c\u65e0\u89e3\u3002<\/li>\n
- \u5f53\u4e09\u4e2a\u5e73\u9762\u76f8\u4ea4\u4e8e\u4e00\u6761\u7ebf\u65f6\uff0c\u65e0\u7a77\u89e3\u3002<\/li>\n
- \u5f53\u4e09\u4e2a\u5e73\u9762\u76f8\u8f83\u4e8e\u4e00\u70b9\u65f6\uff0c\u53ea\u6709\u4e00\u4e2a\u89e3\u3002<\/li>\n<\/ol>\n
import numpy as np\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\n\n# \u5b9a\u4e49\u4e09\u4e2a\u5e73\u9762\ndef plane1(x, y):\n return (7 - x - 2*y) \/ 3\n\ndef plane2(x, y):\n return (4 - 2*x + y) \/ 2\n\ndef plane3(x, y):\n return 3 - x - y\n\n# \u5b9a\u4e49x, y\u8303\u56f4\nx = np.linspace(-2, 2, 400)\ny = np.linspace(-2, 2, 400)\nX, Y = np.meshgrid(x, y)\n\n# \u8ba1\u7b97z\u503c\nZ1 = plane1(X, Y)\nZ2 = plane2(X, Y)\nZ3 = plane3(X, Y)\n\n# \u521b\u5efa\u56fe\u5f62\u5bf9\u8c61\nfig = plt.figure(figsize=(18, 6))\n\n# \u7b2c\u4e00\u79cd\u60c5\u51b5\uff1a\u65e0\u89e3\nax1 = fig.add_subplot(131, projection='3d')\nax1.plot_surface(X, Y, Z1, alpha=0.5, rstride=100, cstride=100)\nax1.plot_surface(X, Y, Z1 + 2, alpha=0.5, rstride=100, cstride=100)\nax1.plot_surface(X, Y, Z1 + 4, alpha=0.5, rstride=100, cstride=100)\nax1.set_title('No Solution')\nax1.set_xlabel('X')\nax1.set_ylabel('Y')\nax1.set_zlabel('Z')\n\n# \u7b2c\u4e8c\u79cd\u60c5\u51b5\uff1a\u65e0\u7a77\u89e3\nax2 = fig.add_subplot(132, projection='3d')\nax2.plot_surface(X, Y, Z1, alpha=0.5, rstride=100, cstride=100)\nax2.plot_surface(X, Y, Z2, alpha=0.5, rstride=100, cstride=100)\nax2.plot_surface(X, Y, Z1 + Z2, alpha=0.5, rstride=100, cstride=100)\n# \u6dfb\u52a0\u4ea4\u7ebf\nintersection_line_x = np.linspace(-2, 2, 100)\nintersection_line_y = intersection_line_x\nintersection_line_z = plane1(intersection_line_x, intersection_line_y)\nax2.plot(intersection_line_x, intersection_line_y, intersection_line_z, color='red', linewidth=2)\nax2.set_title('Infinite Solutions')\nax2.set_xlabel('X')\nax2.set_ylabel('Y')\nax2.set_zlabel('Z')\n\n# \u7b2c\u4e09\u79cd\u60c5\u51b5\uff1a\u4e00\u4e2a\u89e3\nax3 = fig.add_subplot(133, projection='3d')\nax3.plot_surface(X, Y, Z1, alpha=0.5, rstride=100, cstride=100)\nax3.plot_surface(X, Y, Z2, alpha=0.5, rstride=100, cstride=100)\nax3.plot_surface(X, Y, Z3, alpha=0.5, rstride=100, cstride=100)\n# \u6dfb\u52a0\u4ea4\u70b9\nintersection_point = np.array([1, 2, 0]) # \u793a\u4f8b\u4ea4\u70b9\nax3.scatter(intersection_point[0], intersection_point[1], intersection_point[2], color='red', s=100)\nax3.set_title('One Solution')\nax3.set_xlabel('X')\nax3.set_ylabel('Y')\nax3.set_zlabel('Z')\n\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_7_0.png\")
\n<\/p>\n
\u518d\u6765\u770b\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u4e00\u822c\u516c\u5f0f:
\n$$
\n\\begin{cases}
\na_{11} x_1+a_{12} x_2+\\cdots+a_{1 n} x_n=b_1 \\\\
\na_{21} x_1+a_{22} x_2+\\cdots+a_{2 n} x_n=b_2 \\\\
\n\\vdots \\\\
\na_{m 1} x_1+a_{m 2} x_2+\\cdots+a_{m n} x_n=b_m
\n\\end{cases}
\n$$<\/p>\n
\u5176\u4e2d $a_{i j}$ \u8868\u793a $i$ \u884c $j$ \u5217\u7684\u7cfb\u6570, $x_n$ \u8868\u793a\u53d8\u91cf\/\u672a\u77e5\u6570, $b_m$ \u8868\u793a\u5e38\u6570.<\/p>\n
![](\"https:\/\/img.icons8.com\/?size=100&id=PMheOVWBzPf4&format=png&color=000000\")
\u589e\u5e7f\u77e9\u9635<\/h3>\n
\n
\u4e0a\u8ff0\u65b9\u7a0b\u7ec4\u4e5f\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u7cfb\u7edf\u8fdb\u884c\u7b80\u5316\uff1a<\/p>\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729172549994.png\")
<\/center><\/p>\n\n- \u628a\u7cfb\u6570\u4ece\u7ebf\u6027\u65b9\u7a0b\u7ec4\u4e2d\u63d0\u53d6\u51fa\u6765\uff0c\u5199\u6210\u7684\u77e9\u9635\u79f0\u4e3a\u7cfb\u6570\u77e9\u9635\u3002<\/li>\n
- \u628a\u5e38\u6570\u9879\u4ece\u7ebf\u6027\u65b9\u7a0b\u7ec4\u4e2d\u63d0\u53d6\u51fa\u6765\uff0c\u5199\u6210\u7684\u77e9\u9635\u79f0\u4e3a\u5e38\u6570\u9879\u77e9\u9635\u3002<\/li>\n
- \u628a\u7cfb\u6570\u77e9\u9635\u548c\u5e38\u6570\u9879\u77e9\u9635\u5de6\u53f3\u62fc\u63a5\u5728\u4e00\u8d77\uff0c\u5199\u51fa\u7684\u77e9\u9635\u79f0\u4e3a\u589e\u5e7f\u77e9\u9635\u3002<\/li>\n<\/ul>\n
\u7279\u522b\u5730, \u82e5 $b_1=b_2=\\cdots=b_n=0$, \u65b9\u7a0b\u7ec4\u53d8\u4e3a:<\/p>\n
$$
\n\\begin{gathered}
\na_{11} x_1+a_{12} x_2+\\cdots+a_{1 n} x_n=0 \\\\
\na_{21} x_1+a_{22} x_2+\\cdots+a_{2 n} x_n=0 \\\\
\n\\vdots \\\\
\na_{m 1} x_1+a_{m 2} x_2+\\cdots+a_{m n} x_n=0
\n\\end{gathered}
\n$$<\/p>\n
\u79f0\u4e0a\u9762\u7ebf\u6027\u65b9\u7a0b\u7ec4\u4e3a\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u4e0e\u5176\u7cfb\u6570\u77e9\u9635\u4e00\u4e00\u5bf9\u5e94\u3002<\/p>\n
\u6211\u4eec\u5229\u7528\u589e\u5e7f\u77e9\u9635\u7684\u65b9\u6cd5\uff0c\u901a\u8fc7\u521d\u7b49\u884c\u53d8\u6362\u6765\u6c42\u89e3\u65b9\u7a0b\u7ec4:
\n$$
\n\\begin{cases}
\n2 x-y=1 \\\\
\nx+2 y=0
\n\\end{cases}
\n$$<\/p>\n
\n- \n
\u9996\u5148\u5c06\u65b9\u7a0b\u7ec4\u5199\u6210\u589e\u5e7f\u77e9\u9635\u7684\u5f62\u5f0f:
\n$$
\n\\left[\\begin{array}{cc|c}
\n2 & -1 & 1 \\\\
\n1 & 2 & 0
\n\\end{array}\\right]
\n$$<\/p>\n<\/li>\n
- \n
\u6211\u4eec\u5c06\u7b2c 1 \u884c\u4e58\u4ee5 $\\frac{1}{2}$ \uff08\u8bb0\u4f5c $R_1 \\leftarrow \\frac{1}{2} R_1$ ):
\n$$
\n\\left[\\begin{array}{cc|c}
\n1 & -\\frac{1}{2} & \\frac{1}{2} \\\\
\n1 & 2 & 0
\n\\end{array}\\right]
\n$$<\/p>\n<\/li>\n
- \n
\u6211\u4eec\u7528\u7b2c 1 \u884c\u6d88\u53bb\u7b2c 2 \u884c\u4e2d\u7684 $x$ (\u8bb0\u4f5c $R_2 \\leftarrow R_2-R_1$ ):
\n$$
\n\\left[\\begin{array}{cc|c}
\n1 & -\\frac{1}{2} & \\frac{1}{2} \\\\
\n0 & \\frac{5}{2} & -\\frac{1}{2}
\n\\end{array}\\right]
\n$$<\/p>\n<\/li>\n
- \n
\u6211\u4eec\u5c06\u7b2c 2 \u884c\u4e58\u4ee5 $\\frac{2}{5}$ (\u8bb0\u4f5c $R_2 \\leftarrow \\frac{2}{5} R_2$ ):
\n$$
\n\\left[\\begin{array}{cc:c}
\n1 & -\\frac{1}{2} & \\frac{1}{2} \\\\
\n0 & 1 & -\\frac{1}{5}
\n\\end{array}\\right]
\n$$<\/p>\n<\/li>\n
- \n
\u6211\u4eec\u7528\u7b2c 2 \u884c\u6d88\u53bb\u7b2c 1 \u884c\u4e2d\u7684 $y$ \uff08\u8bb0\u4f5c $R_1 \\leftarrow R_1+\\frac{1}{2} R_2$ ):
\n$$
\n\\left[\\begin{array}{cc|c}
\n1 & 0 & \\frac{2}{5} \\\\
\n0 & 1 & -\\frac{1}{5}
\n\\end{array}\\right]
\n$$<\/p>\n<\/li>\n<\/ol>\n
\u6700\u7ec8\u7684\u589e\u5e7f\u77e9\u9635\u8868\u793a\u89e3\u4e3a:
\n$$
\n\\left[\\begin{array}{cc|c}
\n1 & 0 & \\frac{2}{5} \\\\
\n0 & 1 & -\\frac{1}{5}
\n\\end{array}\\right]
\n$$<\/p>\n
\u56e0\u6b64\uff0c\u89e3\u4e3a:
\n$$
\nx=\\frac{2}{5}=0.4, \\quad y=-\\frac{1}{5}=-0.2
\n$$<\/p>\n
\u8fd9\u91cc\u4ec5\u4ee5\u589e\u5e7f\u77e9\u9635\u4e3a\u4f8b\uff0c\u7b80\u5355\u8bf4\u660e\u65b9\u7a0b\u7ec4\u7cfb\u7edf\u548c\u77e9\u9635\u7cfb\u7edf\u662f\u53ef\u4ee5\u5173\u8054\u7684\u3002\u5173\u4e8e\u77e9\u9635\u8868\u793a\uff0c\u4e0b\u6587\u8fd8\u4f1a\u6709\u66f4\u52a0\u8be6\u7ec6\u7684\u8bb2\u89e3\u8bf4\u660e\u3002<\/p>\n
![](\"https:\/\/img.icons8.com\/?size=100&id=H4HJojwANkv1&format=png&color=000000\")
\u5411\u91cf\u8868\u793a<\/h2>\n
\n
\n
\u9996\u5148\u770b\u4e00\u4e0b\u5411\u91cf\u7684\u901a\u4fd7\u7406\u89e3\uff1a \u5411\u91cf\u662f\u4e00\u4e2a\u6307\u4ee4\uff0c\u4e0d\u662f\u4e00\u4e2a\u5750\u6807\uff0c\u53ef\u4ee5\u5b58\u5728\u4e8e\u5750\u6807\u7cfb\u4e0b\u7684\u4efb\u4f55\u4f4d\u7f6e\u3002<\/p>\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729172716215.png\")
<\/center><\/p>\n\u5411\u91cf $\\binom{2}{1}$ \u53ef\u4ee5\u770b\u4f5c\u5411\u53f3\u8d70\u4e24\u4e2a\u5355\u4f4d, \u5411\u4e0a\u8d70\u4e00\u4e2a\u5355\u4f4d\u3002\u5b83\u53ef\u4ee5\u5b58\u5728\u4e8e\u5750\u6807\u7cfb\u4e0b\u7684\u4efb\u4f55\u4f4d\u7f6e\u3002 $\\binom{2}{1}$ \u5e76\u4e0d\u4ee3\u8868\u5176\u5728\u5750\u6807\u7cfb\u4e2d\u7684 $x$ \u8f74\u548c $y$ \u8f74\u5750\u6807\u3002<\/p>\n
\u5411\u91cf\u7684\u6570\u4e58<\/strong>\uff1a<\/p>\n\u5411\u91cf\u7684\u6570\u4e58(scalor)\u6307\u7528\u4e00\u4e2a\u6807\u91cf\u6765\u4e58\u5411\u91cf, \u6539\u53d8\u7684\u662f\u5411\u91cf\u957f\u77ed, \u4e0d\u6539\u53d8\u65b9\u5411\u3002\u5982:<\/p>\n
$$
\n2\\left[\\begin{array}{l}
\n2 \\\\
\n1
\n\\end{array}\\right]=\\left[\\begin{array}{l}
\n4 \\\\
\n2
\n\\end{array}\\right]
\n$$<\/p>\n
\u5411\u91cf\u7684\u52a0\u6cd5<\/strong>\uff1a<\/p>\n\u5411\u91cf\u7684\u52a0\u6cd5\uff08vector addition\uff09\u8ba1\u7b97\u91c7\u7528\u5e73\u884c\u56db\u8fb9\u5f62\u6cd5\u5219\uff08\u9996\u5c3e\u76f8\u8fde\uff09\uff1a\u4ee5\u540c\u4e00\u8d77\u70b9\u7684\u4e24\u4e2a\u5411\u91cf\u4e3a\u90bb\u8fb9\u4f5c\u5e73\u884c\u56db\u8fb9\u5f62, \u5219\u4ee5\u516c\u5171\u8d77\u70b9\u4e3a\u8d77\u70b9\u7684\u5bf9\u89d2\u7ebf\u6240\u5bf9\u5e94\u5411\u91cf\u5c31\u662f\u548c\u5411\u91cf\u3002\u5982\u4e0b\u56fe\uff1a<\/p>\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729172753676.png\")
<\/center><\/p>\n\u4e3e\u4e2a\u4f8b\u5b50:
\n$$
\n\\left[\\begin{array}{l}
\n2 \\\\
\n1
\n\\end{array}\\right]+\\left[\\begin{array}{c}
\n-1 \\\\
\n1
\n\\end{array}\\right]=\\left[\\begin{array}{l}
\n1 \\\\
\n2
\n\\end{array}\\right]
\n$$<\/p>\n
\u6309\u7167\u6307\u4ee4\u7ffb\u8bd1\u7684\u8bdd: \u5411\u53f3\u79fb\u52a8\u4e24\u4e2a\u5355\u4f4d $\\rightarrow$ \u5411\u4e0a\u79fb\u52a8\u4e00\u4e2a\u5355\u4f4d $\\rightarrow$ \u5411\u5de6\u79fb\u52a8\u4e00\u4e2a\u5355\u4f4d $\\rightarrow$ \u5411\u4e0a\u79fb\u52a8\u4e00\u4e2a\u5355\u4f4d\u3002<\/p>\n
\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408<\/strong>\uff1a<\/p>\n\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff08linear combination\uff09\u5b9e\u9645\u4e0a\u5c31\u662f\u5411\u91cf\u6570\u4e58\u548c\u5411\u91cf\u52a0\u6cd5\u7684\u7ec4\u5408\u3002\u53ef\u4ee5\u7528\u516c\u5f0f $x_1 a_1+x_2 a_2+\\cdots+x_n a_n$ \u8868\u793a\u3002\u5176\u4e2d $x_n$ \u662f\u5e38\u6570, \u5982:<\/p>\n
$$
\n2\\left[\\begin{array}{l}
\n2 \\\\
\n1
\n\\end{array}\\right]-\\left[\\begin{array}{c}
\n-1 \\\\
\n1
\n\\end{array}\\right]=\\left[\\begin{array}{l}
\n5 \\\\
\n1
\n\\end{array}\\right]
\n$$<\/p>\n
\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u4e0e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7d27\u5bc6\u76f8\u5173\uff1a\u5f53\u5bfb\u6c42\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u7684\u89e3\u65f6, \u5b9e\u9645\u4e0a\u662f\u5728\u89e3\u51b3\u4e00\u4e2a\u5bf9\u5e94\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4, \u5982:<\/p>\n
\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/\u5fae\u4fe1\u622a\u56fe_20240725204648.jpg\")
\n<\/p>\n
\u5411\u91cf\u7a7a\u95f4<\/strong>\uff1a<\/p>\n\u5411\u91cf\u7a7a\u95f4\u6307\u7684\u662f\u7ebf\u6027\u7ec4\u5408\u7684\u96c6\u5408\uff0c\u4f8b\u5982 :
\n$$
\n\\boldsymbol{b}=x_1\\left[\\begin{array}{l}
\n2 \\\\
\n1
\n\\end{array}\\right]+x_2\\left[\\begin{array}{c}
\n-1 \\\\
\n1
\n\\end{array}\\right]
\n$$<\/p>\n
$\\boldsymbol{b}$ \u7684\u5411\u91cf\u7a7a\u95f4\u662f\u6574\u4e2a\u4e8c\u7ef4\u7a7a\u95f4\u3002\u5373: \u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4efb\u4f55\u4e00\u4e2a\u5411\u91cf $\\boldsymbol{b}$, \u90fd\u53ef\u4ee5\u901a\u8fc7\u5411\u91cf $\\left[\\begin{array}{l}2 \\\\ 1\\end{array}\\right]$ \u548c $\\left[\\begin{array}{c}-1 \\\\ 1\\end{array}\\right]$ \u7684\u7ebf\u6027\u7ec4\u5408\u8fdb\u884c\u8868\u793a\u3002\u5411\u91cf\u7a7a\u95f4\u7684\u4e25\u8c28\u5b9a\u4e49\u662f\uff1a\u5bf9\u5411\u91cf\u52a0\u6cd5\u548c\u6570\u4e58 (\u5373\u7ebf\u6027\u7ec4\u5408) \u90fd\u5c01\u95ed\u7684\u975e\u7a7a\u96c6\u5408\uff0c\u5c31\u662f\u5411\u91cf\u7a7a\u95f4\u3002<\/p>\n
\u600e\u4e48\u786e\u5b9a\u4e00\u4e2a\u5411\u91cf $\\boldsymbol{b}$ \u662f\u5426\u5728 {$ {\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\cdots \\boldsymbol{a}_n}$ }\u7684\u5411\u91cf\u7a7a\u95f4\u4e2d\u5462? <\/p>\n
\u5176\u5b9e\u5c31\u662f\u53bb\u6c42\u89e3\u5411\u91cf $\\boldsymbol{b}$ \u662f\u5426\u53ef\u4ee5\u5728\u5411\u91cf\u7a7a\u95f4\u4e2d\u88ab\u8868\u793a\u3002<\/p>\n
\u5982\u5224\u65ad\u5411\u91cf $\\left[\\begin{array}{c}-1 \\\\ 4 \\\\ 11\\end{array}\\right]$ \u662f\u5426\u5b58\u5728\u4e8e\u5411\u91cf\u7a7a\u95f4 ![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729174218633.png\")
\n<\/p>\n<\/p>\n
\u53ef\u4ee5\u628a\u8fd9\u4e2a\u95ee\u9898\u8f6c\u5316\u6210\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u7684\u95ee\u9898\u3002\u5373\u6c42\u89e3\u4e00\u4e2a\u5411\u91cf\u662f\u5426\u5728\u5411\u91cf\u7a7a\u95f4\u4e2d, \u5c31\u662f\u6c42\u5411\u91cf\u5bf9\u5e94\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u662f\u5426\u6709\u89e3\u3002\u5176\u8f6c\u5316\u6210\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u8fc7\u7a0b\u5982\u4e0b\u6240\u793a:<\/p>\n
$$
\n\\begin{gathered}
\nx_1\\left[\\begin{array}{r}
\n1 \\\\
\n2 \\\\
\n-4
\n\\end{array}\\right]+x_2\\left[\\begin{array}{c}
\n-3 \\\\
\n-5 \\\\
\n13
\n\\end{array}\\right]+x_3\\left[\\begin{array}{r}
\n2 \\\\
\n-1 \\\\
\n-12
\n\\end{array}\\right]=\\left[\\begin{array}{c}
\n-1 \\\\
\n4 \\\\
\n11
\n\\end{array}\\right] \\\\
\n{\\left[\\begin{array}{c}
\nx_1 \\\\
\n2 x_1 \\\\
\n-4 x_1
\n\\end{array}\\right]+\\left[\\begin{array}{c}
\n-3 x_2 \\\\
\n-5 x_2 \\\\
\n13 x_2
\n\\end{array}\\right]+\\left[\\begin{array}{c}
\n2 x_3 \\\\
\n-x_3 \\\\
\n-12 x_3
\n\\end{array}\\right]=\\left[\\begin{array}{c}
\n-1 \\\\
\n4 \\\\
\n11
\n\\end{array}\\right]} \\\\
\n{\\left[\\begin{array}{c}
\nx_1-3 x_2+2 x_3 \\\\
\n2 x_1-5 x_2-x_3 \\\\
\n-4 x_1+13 x_2-12 x_3
\n\\end{array}\\right]=\\left[\\begin{array}{c}
\n-1 \\\\
\n4 \\\\
\n11
\n\\end{array}\\right]} \\\\
\n{\\left[\\begin{array}{ccc}
\n1 & -3 & 2 \\\\
\n2 & -5 & -1 \\\\
\n-4 & 13 & -12
\n\\end{array}\\right]\\left[\\begin{array}{l}
\nx_1 \\\\
\nx_2 \\\\
\nx_3
\n\\end{array}\\right]=\\left[\\begin{array}{c}
\n-1 \\\\
\n4 \\\\
\n11
\n\\end{array}\\right]}
\n\\end{gathered}
\n$$<\/p>\n
\u6c42\u89e3\u7684\u8bdd\u53ef\u4ee5\u8f6c\u6362\u6210\u589e\u5e7f\u77e9\u9635\u518d\u8fdb\u884c\u6c42\u89e3\u3002\u6700\u540e\u53ef\u89e3\u5f97\uff1a<\/p>\n
$$
\n\\begin{aligned}
\n& x_1=30 \\\\
\n& x_2=11 \\\\
\n& x_3=1
\n\\end{aligned} \\Rightarrow \\boldsymbol{x}=\\left[\\begin{array}{c}
\n30 \\\\
\n11 \\\\
\n1
\n\\end{array}\\right]
\n$$<\/p>\n
\u7ebf\u6027\u76f8\u5173\u548c\u7ebf\u6027\u65e0\u5173<\/strong><\/p>\n\u5982\u679c $x_1 \\boldsymbol{a}_1+x_2 \\boldsymbol{a}_2+\\cdots+x_n \\boldsymbol{a}_n=\\mathbf{0}$, \u53ef\u4ee5\u627e\u5230\u81f3\u5c11\u4e00\u4e2a $x_i$ \u4e0d\u4e3a 0 , \u5373 $x_1, x_2, \\cdots, x_n$ \u4e0d\u5168\u4e3a 0 ,\u5219 {$ {\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\cdots \\boldsymbol{a}_n}$ } \u7ebf\u6027\u76f8\u5173\u3002<\/p>\n
\u5982\u679c $x_1 \\boldsymbol{a}_1+x_2 \\boldsymbol{a}_2+\\cdots+x_n \\boldsymbol{a}_n=\\mathbf{0}$, \u53ea\u5728 $x_1=x_2=\\cdots=x_n=0$ \u7684\u60c5\u51b5\u4e0b\u6210\u7acb, \u5219 {$ {\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\cdots \\boldsymbol{a}_n}$ }\u7ebf\u6027\u65e0\u5173\u3002<\/p>\n
\u5173\u4e8e\u7ebf\u6027\u76f8\u5173\u6027\u5b58\u5728\u4e00\u4e2a\u5b9a\u7406: $n+1$ \u4e2a $n$ \u7ef4\u5411\u91cf\u5fc5\u7ebf\u6027\u76f8\u5173\u3002\u4f8b\u5982\u4e09\u4e2a 3 \u7ef4\u5411\u91cf\u53ef\u4ee5\u7ebf\u6027\u65e0\u5173, \u4f46\u4e09\u4e2a 2 \u7ef4\u5411\u91cf\u4e00\u5b9a\u7ebf\u6027\u76f8\u5173\u3002<\/p>\n
import matplotlib.pyplot as plt\nimport numpy as np\n\n# \u5b9a\u4e49\u5411\u91cf\nv1 = np.array([0, 3])\nv2 = np.array([3, 0])\nv3 = np.array([3, 3]) # \u8fd9\u91cc v3 \u662f v1 \u548c v2 \u7684\u7ebf\u6027\u7ec4\u5408\n\n# \u7ed8\u5236\u5411\u91cf\nfig, ax = plt.subplots()\nax.quiver(0, 0, v1[0], v1[1], angles='xy', scale_units='xy', scale=1, color='r', label='v1')\nax.quiver(0, 0, v2[0], v2[1], angles='xy', scale_units='xy', scale=1, color='g', label='v2')\nax.quiver(0, 0, v3[0], v3[1], angles='xy', scale_units='xy', scale=1, color='b', label='v3 = c1*v1 + c2*v2')\n\n# \u8bbe\u7f6e\u56fe\u5f62\u5c5e\u6027\nax.set_xlim(-1, 4)\nax.set_ylim(-1, 4)\nax.set_aspect('equal')\nplt.grid(True)\nplt.axhline(0, color='black',linewidth=0.5)\nplt.axvline(0, color='black',linewidth=0.5)\nplt.legend()\n\n# \u663e\u793a\u56fe\u5f62\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_18_0.png\")
\n<\/p>\n
from mpl_toolkits.mplot3d import Axes3D\n\n# Define vectors\nv1 = np.array([0, 0, 3])\nv2 = np.array([0, 3, 0])\nv3 = np.array([3, 0, 0])\nv4 = np.array([6, 7, 6]) # v4 is a linear combination of v1, v2, and v3\n\n# Create 3D plot\nfig = plt.figure()\nax = fig.add_subplot(111, projection='3d')\n\n# Plotting vectors\nax.quiver(0, 0, 0, v1[0], v1[1], v1[2], color='r', label='$\\mathbf{v_1}$')\nax.quiver(0, 0, 0, v2[0], v2[1], v2[2], color='g', label='$\\mathbf{v_2}$')\nax.quiver(0, 0, 0, v3[0], v3[1], v3[2], color='b', label='$\\mathbf{v_3}$')\nax.quiver(0, 0, 0, v4[0], v4[1], v4[2], color='y', label='$\\mathbf{v_4}$')\n\n# Setting plot attributes\nax.set_xlim([-1, 7])\nax.set_ylim([-1, 7])\nax.set_zlim([-1, 7])\nax.set_xlabel('X')\nax.set_ylabel('Y')\nax.set_zlabel('Z') \nplt.legend()\n\n# Show plot\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_19_0-1.png\")
\n<\/p>\n
\u5411\u91cf\u7684\u70b9\u79ef\u548c\u5185\u79ef\uff08Inner Product, dot product\uff09<\/strong>\uff1a<\/p>\n\u5411\u91cf\u7684\u70b9\u79ef\u548c\u5185\u79ef (Inner Product, dot product), \u7528\u30fb\u8868\u793a, \u4e24\u4e2a\u5411\u91cf\u7684\u884c\u5217\u6570\u5fc5\u987b\u76f8\u540c, \u70b9\u79ef\u7684\u7ed3\u679c\u662f\u5bf9\u5e94\u5143\u7d20\u76f8\u4e58\u540e\u6c42\u548c, \u7ed3\u679c\u662f\u4e00\u4e2a\u6807\u91cf, \u5982:
\n$$
\n\\begin{aligned}
\n& \\boldsymbol{a}=\\left(a_1, a_2, \\cdots, a_n\\right) \\\\
\n& \\boldsymbol{b}=\\left(b_1, b_2, \\cdots, b_n\\right) \\\\
\n& \\boldsymbol{a} \\cdot \\boldsymbol{b}=a_1 b_1+a_2 b_2+\\cdots+a_n b_n
\n\\end{aligned}
\n$$<\/p>\n
\u70b9\u79ef\u7684\u51e0\u4f55\u610f\u4e49\u53ef\u4ee5\u7528\u6765\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u5939\u89d2: $\\cos \\theta=\\frac{\\boldsymbol{a} \\boldsymbol{b}}{|\\boldsymbol{a}||\\boldsymbol{b}|}$<\/p>\n
\u81f3\u4e8e\u5411\u91cf\u957f\u5ea6\u7684\u6c42\u89e3, $n$ \u7ef4\u5411\u91cf\u7684\u957f\u5ea6: $|\\boldsymbol{x}|=\\sqrt{[x, x]}=\\sqrt{x_1^2+x_2^2+\\cdots+x_n^2} \\geq 0$, \u5f53 $|x|=1$ \u65f6\u79f0\u4e3a\u5355\u4f4d\u5411\u91cf\u3002<\/p>\n
tips<\/strong>\uff1a Dot product \u548c Inner product\u5176\u5b9e\u8fd8\u662f\u6709\u533a\u522b\u7684\uff0c\u70b9\u79ef\u4e3b\u8981\u5e94\u7528\u4e8e\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\uff08\u5373\u666e\u901a\u7684\u4e8c\u7ef4\u6216\u4e09\u7ef4\u7a7a\u95f4\uff09\uff1b\u800c\u5185\u79ef\u662f\u70b9\u79ef\u7684\u4e00\u4e2a\u5e7f\u4e49\u6982\u5ff5\uff0c\u5b83\u4e0d\u4ec5\u4ec5\u5c40\u9650\u4e8e\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\uff0c\u8fd8\u53ef\u4ee5\u5e94\u7528\u4e8e\u66f4\u5e7f\u6cdb\u7684\u5411\u91cf\u7a7a\u95f4\uff0c\u4f8b\u5982\u590d\u6570\u7a7a\u95f4\u6216\u66f4\u62bd\u8c61\u7684\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u3002\u76ee\u524d\u9636\u6bb5\u6682\u65f6\u5c06\u4e8c\u8005\u89c6\u4e3a\u540c\u4e00\u4e2a\u6982\u5ff5\u5373\u53ef\u3002<\/p>\n\u5411\u91cf\u7684\u5916\u79ef<\/strong>\uff1a<\/p>\n\u5411\u91cf\u7684\u5916\u79ef (Outer product), \u7528 $\\otimes$ \u8868\u793a, \u5982:
\n$$
\n\\begin{aligned}
\n& \\boldsymbol{u}=\\left(u_1, u_2, \\cdots, u_m\\right) \\\\
\n& \\boldsymbol{v}=\\left(v_1, v_2, \\cdots, v_n\\right) \\\\
\n& \\boldsymbol{u} \\otimes \\boldsymbol{v}=\\left[\\begin{array}{cccc}
\nu_1 v_1 & u_1 v_2 & \\cdots & u_1 v_n \\\\
\nu_2 v_1 & u_2 v_2 & \\cdots & u_2 v_n \\\\
\n\\vdots & \\vdots & \\ddots & \\vdots \\\\
\nu_m v_1 & u_m v_2 & \\cdots & u_m v_n
\n\\end{array}\\right]
\n\\end{aligned}
\n$$<\/p>\n
\u5411\u91cf\u7684\u53c9\u79ef<\/strong>\uff1a<\/p>\n\u5411\u91cf\u7684\u53c9\u79ef (Outer product), \u7528 $\\times$ \u8868\u793a, \u5982:
\n$$
\n\\begin{aligned}
\n& \\boldsymbol{a}=\\left(x_1, y_1, z_1\\right) \\\\
\n& \\boldsymbol{b}=\\left(x_2, y_2, z_2\\right) \\\\
\n& \\boldsymbol{a} \\times \\boldsymbol{b}=\\left|\\begin{array}{lll}
\n\\boldsymbol{i} & \\boldsymbol{j} & \\boldsymbol{k} \\\\
\nx_1 & y_1 & z_1 \\\\
\nx_2 & y_2 & z_2
\n\\end{array}\\right|=\\left(y_1 z_2-y_2 z_1\\right) \\boldsymbol{i}+\\left(z_1 x_2-z_2 x_1\\right) \\boldsymbol{j}+\\left(x_1 y_2-x_2 y_1\\right) \\boldsymbol{k} \\\\
\n& \\boldsymbol{i}=[1,0,0], \\boldsymbol{j}=[0,1,0], \\boldsymbol{k}=[0,0,1]
\n\\end{aligned}
\n$$<\/p>\n
\u53c9\u79ef\u8ba1\u7b97\u7ed3\u679c\u7684\u51e0\u4f55\u610f\u4e49\u662f\u4e24\u4e2a\u5411\u91cf\u7684\u6cd5\u5411\u91cf\u3002<\/p>\n
\u4e3e\u4e2a\u4f8b\u5b50: $\\boldsymbol{a}$\u662f $x$ \u8f74\u7684\u5355\u4f4d\u5411\u91cf, $\\boldsymbol{b}$ \u662f $y$ \u8f74\u7684\u5355\u4f4d\u5411\u91cf, \u4e8c\u8005\u53c9\u79ef\u7684\u7ed3\u679c\u5c31\u662f $z$ \u8f74\u7684\u5355\u4f4d\u5411\u91cf, \u5373:<\/p>\n
$$
\n\\begin{aligned}
\n& \\boldsymbol{a}=(1,0,0) \\\\
\n& \\boldsymbol{b}=(0,1,0) \\\\
\n& \\boldsymbol{i}=(1,0,0) \\\\
\n& \\boldsymbol{j}=(0,1,0) \\\\
\n& \\boldsymbol{k}=(0,0,1) \\\\
\n& \\boldsymbol{a} \\times \\boldsymbol{b}=\\left|\\begin{array}{lll}
\n\\boldsymbol{i} & \\boldsymbol{j} & \\boldsymbol{k} \\\\
\n1 & 0 & 0 \\\\
\n0 & 1 & 0
\n\\end{array}\\right|=(0 \\times 0-0 \\times 1) \\boldsymbol{i}+(0 \\times 0-0 \\times 1) \\boldsymbol{j}+(1 \\times 1-0 \\times 0) \\boldsymbol{k}=\\boldsymbol{k}
\n\\end{aligned}
\n$$<\/p>\n
\u6b63\u4ea4\u4e0e\u57fa\u7ec4<\/strong><\/p>\n\u6b63\u4ea4<\/strong>\u662f\u6307\u4e24\u4e2a\u5411\u91cf\u7684\u5185\u79ef\u4e3a\u96f6\u7684\u60c5\u51b5\u3002\u5728\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\uff0c\u5982\u679c\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\u4e3a\u96f6\uff0c\u5b83\u4eec\u88ab\u79f0\u4e3a\u6b63\u4ea4\u5411\u91cf\u3002<\/p>\n\u5bf9\u4e8e\u4e24\u4e2a\u5411\u91cf $\\mathbf{u}$ \u548c $\\mathbf{v}$ \uff0c\u5982\u679c:
\n$$
\n\\mathbf{u} \\cdot \\mathbf{v}=0
\n$$<\/p>\n
\u90a3\u4e48\u8fd9\u4e24\u4e2a\u5411\u91cf\u5c31\u662f\u6b63\u4ea4\u7684\u3002\u6b63\u4ea4\u5411\u91cf\u5177\u6709\u4e00\u4e9b\u91cd\u8981\u6027\u8d28:<\/p>\n
\n- \u72ec\u7acb\u6027: \u6b63\u4ea4\u5411\u91cf\u4e00\u5b9a\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<\/li>\n
- \u957f\u5ea6\u4e0d\u53d8\u6027: \u6b63\u4ea4\u5411\u91cf\u7684\u957f\u5ea6\u5728\u5185\u79ef\u8fd0\u7b97\u4e2d\u4fdd\u6301\u4e0d\u53d8\u3002<\/li>\n
- \u6295\u5f71: \u4efb\u610f\u5411\u91cf\u5728\u6b63\u4ea4\u5411\u91cf\u4e0a\u7684\u6295\u5f71\u8ba1\u7b97\u975e\u5e38\u7b80\u5355\u3002<\/li>\n<\/ol>\n
\u57fa\u7ec4<\/strong>\u662f\u6307\u5411\u91cf\u7a7a\u95f4\u4e2d\u4e00\u7ec4\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\uff0c\u8fd9\u7ec4\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u53ef\u4ee5\u8868\u793a\u8be5\u7a7a\u95f4\u4e2d\u7684\u4efb\u610f\u5411\u91cf\u3002\u57fa\u7ec4\u7684\u4e2a\u6570\u7b49\u4e8e\u5411\u91cf\u7a7a\u95f4\u7684\u7ef4\u6570\u3002<\/p>\n\u8bbe $V$ \u662f\u4e00\u4e2a $n$ \u7ef4\u5411\u91cf\u7a7a\u95f4\uff0c\u5982\u679c\u5411\u91cf\u96c6\u5408 {$ {\\boldsymbol{v}_1,\\boldsymbol{v}_2,\\cdots \\boldsymbol{v}_n}$ }\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6:<\/p>\n
\n- \u7ebf\u6027\u65e0\u5173\uff1a\u96c6\u5408\u4e2d\u7684\u5411\u91cf\u5f7c\u6b64\u7ebf\u6027\u65e0\u5173\u3002<\/li>\n
- \u751f\u6210\u6574\u4e2a\u7a7a\u95f4: \u96c6\u5408\u4e2d\u7684\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u53ef\u4ee5\u751f\u6210 $V$ \u4e2d\u7684\u4efb\u610f\u5411\u91cf\u3002<\/li>\n<\/ol>\n
\u90a3\u4e48\uff0c\u96c6\u5408 {$ {\\boldsymbol{v}_1,\\boldsymbol{v}_2,\\cdots \\boldsymbol{v}_n}$ } \u5c31\u662f $V$ \u7684\u4e00\u4e2a\u57fa\u7ec4\u3002\u57fa\u7ec4\u7684\u6982\u5ff5\u5728\u6570\u5b66\u4e2d\u662f\u975e\u5e38\u57fa\u7840\u548c\u91cd\u8981\u7684\uff0c\u5b83\u63d0\u4f9b\u4e86\u4e00\u79cd\u5bf9\u5411\u91cf\u7a7a\u95f4\u8fdb\u884c\u63cf\u8ff0\u548c\u64cd\u4f5c\u7684\u65b9\u6cd5\u3002<\/p>\n
\u5728\u8bb8\u591a\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u66f4\u5e0c\u671b\u4f7f\u7528\u6b63\u4ea4\u57fa\u6216\u6807\u51c6\u6b63\u4ea4\u57fa:<\/p>\n
\n- \u6b63\u4ea4\u57fa\uff1a\u5982\u679c\u57fa\u7ec4\u4e2d\u7684\u5411\u91cf\u4e24\u4e24\u6b63\u4ea4\uff0c\u5219\u79f0\u4e3a\u6b63\u4ea4\u57fa\u3002<\/li>\n
- \u6807\u51c6\u6b63\u4ea4\u57fa\uff1a\u5982\u679c\u57fa\u7ec4\u4e2d\u7684\u5411\u91cf\u4e24\u4e24\u6b63\u4ea4\u4e14\u6bcf\u4e2a\u5411\u91cf\u7684\u957f\u5ea6\u4e3a1\uff0c\u5219\u79f0\u4e3a\u6807\u51c6\u6b63\u4ea4\u57fa\u3002<\/li>\n<\/ul>\n
\u6807\u51c6\u6b63\u4ea4\u57fa\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9:<\/p>\n
\n- \u7b80\u5316\u8ba1\u7b97: \u5411\u91cf\u7684\u8868\u793a\u548c\u8fd0\u7b97\uff08\u5982\u5185\u79ef\u3001\u6295\u5f71\uff09\u53d8\u5f97\u66f4\u52a0\u7b80\u5355\u3002<\/li>\n
- \u7a33\u5b9a\u6027\uff1a\u5728\u6570\u503c\u8ba1\u7b97\u4e2d\uff0c\u6807\u51c6\u6b63\u4ea4\u57fa\u53ef\u4ee5\u51cf\u5c11\u8bef\u5dee\u7d2f\u79ef\u3002<\/li>\n<\/ol>\n
![](\"https:\/\/img.icons8.com\/?size=100&id=BgtYnJlfI2-j&format=png&color=000000\")
\u77e9\u9635\u8868\u793a<\/h3>\n
\n
\n
\u77e9\u9635\u4e58\u6cd5<\/strong><\/p>\n\u77e9\u9635\u4e58\u6cd5 (Matmul Product) \u662f\u4e24\u4e2a\u77e9\u5f62\u76f8\u4e58\u7684\u64cd\u4f5c, \u5176\u7ed3\u679c\u662f\u53e6\u4e00\u4e2a\u77e9\u9635\u3002\u5b9a\u4e49\u5982\u4e0b:\u8bbe\u6709\u4e24\u4e2a\u77e9\u9635 $\\boldsymbol{A}$ \u548c $\\boldsymbol{B}$, \u4ee4 $\\boldsymbol{A}$ \u662f\u4e00\u4e2a $m \\times n$ \u7684\u77e9\u9635, \u800c $\\boldsymbol{B}$ \u662f\u4e00\u4e2a $n \\times p$ \u7684\u77e9\u9635\u3002\u90a3\u4e48\u77e9\u9635 $\\boldsymbol{A}$\u548c $\\boldsymbol{B}$ \u7684\u4e58\u79ef $\\boldsymbol{C}$ \u662f\u4e00\u4e2a $m \\times p$ \u7684\u77e9\u9635, \u6bcf\u4e2a\u5143\u7d20\u7531\u4ee5\u4e0b\u516c\u5f0f\u7ed9\u51fa:
\n$$
\n\\boldsymbol{C}_{i j}=\\sum_{k=1}^n \\boldsymbol{A}_{i k} \\boldsymbol{B}_{k j}
\n$$<\/p>\n
\u5176\u4e2d $\\boldsymbol{C}_{i j}$ \u662f\u7ed3\u679c\u77e9\u9635 $\\boldsymbol{C}$ \u7684\u7b2c $i$ \u884c\u7b2c $j$ \u5217\u7684\u5143\u7d20\u3002<\/p>\n
\u8bbe\u6709\u77e9\u9635 $\\boldsymbol{A}$ \u548c $\\boldsymbol{B}$ \u5982\u4e0b:
\n$$
\n\\begin{aligned}
\n\\boldsymbol{A} & =\\left[\\begin{array}{ll}
\n1 & 2 \\\\
\n3 & 4
\n\\end{array}\\right] \\\\
\n\\boldsymbol{B} & =\\left[\\begin{array}{ll}
\n2 & 0 \\\\
\n1 & 3
\n\\end{array}\\right]
\n\\end{aligned}
\n$$<\/p>\n
\u8ba1\u7b97 $\\boldsymbol{A} \\times \\boldsymbol{B}$ \u7684\u7ed3\u679c\u4e3a:
\n$$
\n\\boldsymbol{A} \\times \\boldsymbol{B}=\\left[\\begin{array}{cc}
\n4 & 6 \\\\
\n10 & 12
\n\\end{array}\\right]
\n$$<\/p>\n
\u54c8\u8fbe\u739b\u79ef<\/strong><\/p>\n\u54c8\u8fbe\u739b\u79ef\uff08Element-wise Product\uff09\u8868\u793a\u4e24\u4e2a\u77e9\u9635\u5bf9\u5e94\u5143\u7d20\u76f8\u4e58, \u4e8c\u8005\u7ef4\u6570\u5fc5\u987b\u76f8\u540c, \u7528 $\\odot$ \u8868\u793a\u3002\u5982\uff1a<\/p>\n
$$
\n\\left[\\begin{array}{ll}
\na_{11} & a_{12} \\\\
\na_{21} & a_{22}
\n\\end{array}\\right] \\odot\\left[\\begin{array}{ll}
\nb_{11} & b_{12} \\\\
\nb_{21} & b_{22}
\n\\end{array}\\right]=\\left[\\begin{array}{ll}
\na_{11} b_{11} & a_{12} b_{12} \\\\
\na_{21} b_{21} & a_{22} b_{22}
\n\\end{array}\\right]
\n$$
\n\u514b\u7f57\u5185\u514b\u79ef<\/strong><\/p>\n\u514b\u7f57\u5185\u514b\u79ef (Kronecker Product) \u8868\u793a\u4e24\u4e2a\u4efb\u610f\u5927\u5c0f\u77e9\u9635\u95f4\u7684\u8fd0\u7b97, \u77e9\u9635 $\\boldsymbol{A}$ \u7684\u6bcf\u4e2a\u5143\u7d20\u9010\u4e2a\u4e0e\u77e9\u9635 $\\boldsymbol{B}$ \u76f8\u4e58, \u7528 $\\otimes$ \u8868\u793a\u3002\u5982:
\n$$
\n\\left[\\begin{array}{ll}
\na_{11} & a_{12} \\\\
\na_{21} & a_{22}
\n\\end{array}\\right] \\otimes\\left[\\begin{array}{ll}
\nb_{11} & b_{12} \\\\
\nb_{21} & b_{22}
\n\\end{array}\\right]=\\left[\\begin{array}{llll}
\na_{11} b_{11} & a_{11} b_{12} & a_{12} b_{11} & a_{12} b_{12} \\\\
\na_{11} b_{21} & a_{11} b_{22} & a_{12} b_{21} & a_{12} b_{22} \\\\
\na_{21} b_{11} & a_{21} b_{12} & a_{22} b_{11} & a_{22} b_{12} \\\\
\na_{21} b_{21} & a_{21} b_{22} & a_{22} b_{21} & a_{22} b_{22}
\n\\end{array}\\right]
\n$$
\n\u521d\u7b49\u77e9\u9635<\/strong><\/p>\n\u77e9\u9635\u53ef\u4ee5\u770b\u4f5c\u5bf9\u5411\u91cf\u7684\u53d8\u6362\uff0c\u5355\u4f4d\u77e9\u9635\u662f\u5bf9\u89d2\u7ebf\u51681\u7684\u77e9\u9635\uff0c\u76f8\u5f53\u4e8e0\u53d8\u6362\u3002<\/p>\n
$$
\n\\left[\\begin{array}{lll}
\n1 & 0 & 0 \\\\
\n0 & 1 & 0 \\\\
\n0 & 0 & 1
\n\\end{array}\\right]=\\boldsymbol{I}
\n$$<\/p>\n
\u53ef\u4ee5\u53d1\u73b0\u521d\u7b49\u77e9\u9635\u662f\u5bf9\u89d2\u77e9\u9635, \u5982\u679c\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\u7684\u503c\u4e0d\u4e3a 1 \u4f1a\u5bf9\u5411\u91cf\u4ea7\u751f\u4ec0\u4e48\u5f71\u54cd\u5462? <\/p>\n
import numpy as np\nimport matplotlib.pyplot as plt\n\n# \u5b9a\u4e49\u539f\u59cb\u5411\u91cf\nvector = np.array([1, 1])\n\n# \u5b9a\u4e49\u5bf9\u89d2\u77e9\u9635\ndiagonal_matrix = np.array([[3, 0], \n [0, 2]])\n\n# \u5bf9\u5411\u91cf\u5e94\u7528\u5bf9\u89d2\u77e9\u9635\u53d8\u6362\ntransformed_vector = diagonal_matrix @ vector\n\n# \u8bbe\u7f6e\u7ed8\u56fe\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))\n\n# \u7b2c\u4e00\u4e2a\u5b50\u56fe\uff1a\u586b\u5145\u53d8\u6362\u524d\u7684\u9634\u5f71\nax1.axhline(0, color='grey', lw=0.5)\nax1.axvline(0, color='grey', lw=0.5)\nax1.grid(True, which='both')\n\n# \u7ed8\u5236\u539f\u59cb\u5411\u91cf\nax1.quiver(0, 0, vector[0], vector[1], angles='xy', scale_units='xy', scale=1, color='blue', label='Original Vector')\n# \u7ed8\u5236\u53d8\u6362\u540e\u7684\u5411\u91cf\nax1.quiver(0, 0, transformed_vector[0], transformed_vector[1], angles='xy', scale_units='xy', scale=1, color='red', label='Transformed Vector')\n\n# \u586b\u5145\u539f\u59cb\u5411\u91cf\u6240\u5728\u7684\u77e9\u5f62\u533a\u57df\nax1.fill([0, vector[0], vector[0], 0], [0, 0, vector[1], vector[1]], color='blue', alpha=0.1)\n\n# \u8bbe\u7f6e\u5750\u6807\u8f74\u8303\u56f4\nax1.set_xlim(-2, 5)\nax1.set_ylim(-2, 5)\n\n# \u8bbe\u7f6e\u5750\u6807\u8f74\u523b\u5ea6\u4e3a1\nax1.set_xticks(np.arange(-2, 6, 1))\nax1.set_yticks(np.arange(-2, 6, 1))\n\n# \u6dfb\u52a0\u56fe\u4f8b\nax1.legend()\nax1.set_title('Original Vector Shadow')\n\n# \u7b2c\u4e8c\u4e2a\u5b50\u56fe\uff1a\u586b\u5145\u53d8\u6362\u540e\u7684\u9634\u5f71\nax2.axhline(0, color='grey', lw=0.5)\nax2.axvline(0, color='grey', lw=0.5)\nax2.grid(True, which='both')\n\n# \u7ed8\u5236\u539f\u59cb\u5411\u91cf\nax2.quiver(0, 0, vector[0], vector[1], angles='xy', scale_units='xy', scale=1, color='blue', label='Original Vector')\n# \u7ed8\u5236\u53d8\u6362\u540e\u7684\u5411\u91cf\nax2.quiver(0, 0, transformed_vector[0], transformed_vector[1], angles='xy', scale_units='xy', scale=1, color='red', label='Transformed Vector')\n\n# \u586b\u5145\u53d8\u6362\u540e\u5411\u91cf\u6240\u5728\u7684\u77e9\u5f62\u533a\u57df\nax2.fill([0, transformed_vector[0], transformed_vector[0], 0], [0, 0, transformed_vector[1], transformed_vector[1]], color='red', alpha=0.1)\n\n# \u8bbe\u7f6e\u5750\u6807\u8f74\u8303\u56f4\nax2.set_xlim(-2, 5)\nax2.set_ylim(-2, 5)\n\n# \u8bbe\u7f6e\u5750\u6807\u8f74\u523b\u5ea6\u4e3a1\nax2.set_xticks(np.arange(-2, 6, 1))\nax2.set_yticks(np.arange(-2, 6, 1))\n\n# \u6dfb\u52a0\u56fe\u4f8b\nax2.legend()\nax2.set_title('Transformed Vector Shadow')\n\n# \u663e\u793a\u56fe\u5f62\nplt.tight_layout()\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_29_0-1.png\")
\n<\/p>\n
\u53ef\u4ee5\u53d1\u73b0\uff0c\u6b64\u65f6\u7684\u5f71\u54cd\u662f\u5728x\u8f74\u6216y\u8f74\u4e0a\u5bf9\u5411\u91cf\u8fdb\u884c\u4f38\u7f29\u3002\u90a3\u4e48\u666e\u901a\u7684\u77e9\u9635\u53c8\u662f\u4ec0\u4e48\u6548\u679c\u5462\uff1f<\/p>\n
\u666e\u901a\u7684\u77e9\u9635\u8fd8\u53ef\u4ee5\u5bf9\u5411\u91cf\u4ea7\u751f\u65cb\u8f6c\u7684\u53d8\u6362\u6548\u679c<\/strong><\/p>\n\u53ef\u9006\u77e9\u9635<\/strong><\/p>\n\u5728\u7ebf\u6027\u4ee3\u6570\u4e2d\uff0c\u53ef\u9006\u77e9\u9635 (Invertible Matrix)\uff0c\u53c8\u79f0\u4e3a\u975e\u5947\u5f02\u77e9\u9635 (Non-Singular Matrix) \u6216\u6ee1\u79e9\u77e9\u9635 (Full-Rank Matrix)\uff0c\u662f\u6307\u5b58\u5728\u552f\u4e00\u9006\u77e9\u9635\u7684\u65b9\u9635\u3002\u4e00\u4e2a $n \\times n$ \u7684\u65b9\u9635 $A$ \u79f0\u4e3a\u53ef\u9006\u7684\uff0c\u5982\u679c\u5b58\u5728\u53e6\u4e00\u4e2a $n \\times n$ \u7684\u77e9\u9635 $B$ \uff0c\u4f7f\u5f97:
\n$$
\nA \\cdot B=B \\cdot A=I
\n$$<\/p>\n
\u5176\u4e2d\uff0c $I$ \u662f $n \\times n$ \u7684\u5355\u4f4d\u77e9\u9635\uff08\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5168\u4e3a 1 \uff0c\u5176\u4f59\u5143\u7d20\u5168\u4e3a 0 \uff09\uff0c\u800c\u77e9\u9635 $B$ \u5c31\u662f\u77e9\u9635 $A$\u7684\u9006\u77e9\u9635\uff0c\u8bb0\u4f5c $A^{-1}$ \u3002<\/p>\n
\u53ef\u9006\u77e9\u9635\u5728\u8bb8\u591a\u6570\u5b66\u548c\u5de5\u7a0b\u9886\u57df\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5305\u62ec\uff1a<\/p>\n
\n- \u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002<\/li>\n
- \u7ebf\u6027\u53d8\u6362\u4e2d\u7684\u53d8\u6362\u77e9\u9635\u3002<\/li>\n
- \u8ba1\u7b97\u77e9\u9635\u51fd\u6570\uff08\u5982\u77e9\u9635\u6307\u6570\uff09\u3002<\/li>\n
- \u6570\u636e\u5206\u6790\u4e2d\u7684\u4e3b\u6210\u5206\u5206\u6790\uff08PCA\uff09\u3002<\/li>\n<\/ul>\n
\u4e3e\u4f8b\uff1a\u53ef\u9006\u77e9\u9635\u628a\u6c42\u89e3\u5411\u91cf$x$\u7684\u95ee\u9898\u8f6c\u6362\u6210\u4e86\u6c42\u53ef\u9006\u77e9\u9635\u672c\u8eab\u7684\u95ee\u9898\uff0c\u5982\u4e0b\u516c\u5f0f\u63a8\u5bfc\u6240\u793a\uff1a<\/p>\n
$$
\n\\begin{aligned}
\n& A x=b \\\\
\n& A^{-1} A x=A^{-1} b \\\\
\n& x=A^{-1} b
\n\\end{aligned}
\n$$<\/p>\n
\u53ef\u9006\u77e9\u9635\u7684\u6c42\u89e3\u4e0e\u77e9\u9635 $\\boldsymbol{A}$ \u548c\u521d\u7b49\u77e9\u9635 $\\boldsymbol{I}$ \u6709\u5173, \u5177\u4f53\u63a8\u5bfc\u5982\u4e0b:
\n\u5047\u8bbe $\\boldsymbol{A}=\\left[\\begin{array}{ccc}1 & -2 & 1 \\\\ -3 & 7 & -6 \\\\ 2 & -3 & 0\\end{array}\\right]$, \u6c42\u53ef\u9006\u77e9\u9635 $\\boldsymbol{A}^{-1}$ \u7684\u8fc7\u7a0b\u5982\u4e0b:
\n$$
\n\\left[\\begin{array}{ccc:ccc}
\n1 & -2 & 1 & 1 & 0 & 0 \\\\
\n-3 & 7 & -6 & 0 & 1 & 0 \\\\
\n2 & -3 & 0 & 0 & 0 & 1
\n\\end{array}\\right] \\Rightarrow\\left[\\begin{array}{lll:lll}
\n1 & 0 & 0 & -18 & -3 & 5 \\\\
\n0 & 1 & 0 & -12 & -2 & 3 \\\\
\n0 & 0 & 1 & -5 & -1 & 1
\n\\end{array}\\right]
\n$$<\/p>\n
\u901a\u8fc7\u77e9\u9635\u7684\u521d\u7b49\u884c\u53d8\u6362\u5c06\u5de6\u8fb9\u77e9\u9635\u63a8\u5bfc\u6210\u53f3\u9762\u7684\u683c\u5f0f\u5373\u5b8c\u6210\u4e86\u53ef\u9006\u77e9\u9635\u7684\u8ba1\u7b97, \u6b64\u65f6\u53ef\u9006\u77e9\u9635: $\\boldsymbol{A}^{-1}=\\left[\\begin{array}{ccc}-18 & -3 & 5 \\\\ -12 & -2 & 3 \\\\ -5 & -1 & 1\\end{array}\\right]$ \u3002<\/p>\n
\u884c\u5217\u5f0f<\/strong><\/p>\n\u8fd9\u91cc\u5c31\u5148\u8c08\u884c\u5217\u5f0f\u7684\u51e0\u4f55\u610f\u4e49\uff0c\u6700\u540e\u518d\u8c08\u884c\u5217\u5f0f\u7684\u8ba1\u7b97\u65b9\u6cd5\u7684\u7531\u6765\u3002<\/p>\n
\u601d\u8003\u4e00\u4e0b\uff0c\u7ecf\u8fc7\u7ebf\u6027\u53d8\u6362\uff08\u77e9\u9635\u76f8\u4e58\uff09\uff0c\u7a7a\u95f4\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u76f8\u5e94\u7684\u9762\u79ef\u4e5f\u4f1a\u53d1\u751f\u53d8\u5316\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u884c\u5217\u5f0f\u7406\u89e3\u4e3a\u7ebf\u6027\u53d8\u6362\u5bf9\u7a7a\u95f4\uff08\u5305\u62ec\u9762\u79ef\u3001\u4f53\u79ef\u7b49\uff09\u7684\u7f29\u653e\u6bd4\u4f8b\u56e0\u5b50\u3002<\/p>\n
\u8003\u8651\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u77e9\u5f62\uff0c\u5176\u9876\u70b9\u5206\u522b\u4e3a $(0,0) \u3001(1,0) \u3001(0,1) \u3001(1,1)$ \u3002\u5982\u679c\u6211\u4eec\u5bf9\u8fd9\u4e2a\u77e9\u5f62\u8fdb\u884c\u7ebf\u6027\u53d8\u6362\uff0c\u5373\u901a\u8fc7\u77e9\u9635 $A$ \u4f5c\u7528\u5728\u5b83\u4e0a\u9762\uff0c\u8fd9\u4e2a\u77e9\u5f62\u4f1a\u53d8\u5f62\u4e3a\u4e00\u4e2a\u5e73\u884c\u56db\u8fb9\u5f62\u3002<\/p>\n
\u5047\u8bbe\u77e9\u9635 $A$ \u4e3a:
\n$$
\nA=\\left(\\begin{array}{ll}
\na & b \\\\
\nc & d
\n\\end{array}\\right)
\n$$<\/p>\n
\u7ecf\u8fc7\u77e9\u9635 $A$ \u7684\u7ebf\u6027\u53d8\u6362\u540e\uff0c\u57fa\u5411\u91cf $(1,0)$ \u548c $(0,1)$ \u5206\u522b\u53d8\u4e3a $(a, c)$ \u548c $(b, d)$ \u3002\u53d8\u6362\u540e\u7684\u5e73\u884c\u56db\u8fb9\u5f62\u7684\u9876\u70b9\u53d8\u4e3a $(0,0) \u3001(a, c) \u3001(b, d) \u3001(a+b, c+d)$ \u3002<\/p>\n
\u6b64\u65f6\uff0c\u5e73\u884c\u56db\u8fb9\u5f62\u7684\u9762\u79ef\u53ef\u4ee5\u901a\u8fc7\u884c\u5217\u5f0f\u8ba1\u7b97:
\n$$
\n\\text { \u9762\u79ef }=|\\operatorname{det}(A)|=|a d-b c|
\n$$<\/p>\n
\u884c\u5217\u5f0f\u7684\u7edd\u5bf9\u503c\u8868\u793a\u7ebf\u6027\u53d8\u6362\u540e\u9762\u79ef\u7684\u6bd4\u4f8b\u56e0\u5b50\u3002\u6b63\u503c\u8868\u793a\u53d8\u6362\u4fdd\u6301\u4e86\u539f\u6765\u7684\u65b9\u5411\uff0c\u8d1f\u503c\u8868\u793a\u53d8\u6362\u7ffb\u8f6c\u4e86\u65b9\u5411\u3002<\/p>\n
import matplotlib.pyplot as plt\nimport numpy as np\n\n# \u5b9a\u4e49\u539f\u59cb\u5355\u4f4d\u6b63\u65b9\u5f62\u7684\u9876\u70b9\nsquare = np.array([[0, 0], [1, 0], [1, 1], [0, 1], [0, 0]])\n\n# \u5b9a\u4e49\u77e9\u9635 A\nA = np.array([[2, 1], [1, 3]])\n\n# \u7ebf\u6027\u53d8\u6362\u540e\u7684\u9876\u70b9\ntransformed_square = square @ A.T\n\n# \u7ed8\u5236\u539f\u59cb\u5355\u4f4d\u6b63\u65b9\u5f62\nplt.plot(square[:, 0], square[:, 1], 'b-', label='orginal')\n\n# \u7ed8\u5236\u7ebf\u6027\u53d8\u6362\u540e\u7684\u5e73\u884c\u56db\u8fb9\u5f62\nplt.plot(transformed_square[:, 0], transformed_square[:, 1], 'r-', label='transformer')\n\n# \u8bbe\u7f6e\u56fe\u5f62\u5c5e\u6027\nplt.axhline(0, color='black',linewidth=0.5)\nplt.axvline(0, color='black',linewidth=0.5)\nplt.grid(True)\nplt.legend() \nplt.xlabel('x')\nplt.ylabel('y')\n\n# \u663e\u793a\u56fe\u5f62\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_33_0.png\")
\n<\/p>\n
\n ![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729174916938.png\")
\n<\/p>\n
\u7531\u4e8e\u6c42\u7684\u662f\u5e73\u884c\u56db\u8fb9\u5f62\u7684\u9762\u79ef, \u4ece\u56fe\u4e0a\u6784\u5efa\u51fa\u6574\u4e2a\u957f\u65b9\u5f62, \u5f97\u5230\u6bcf\u4e00\u90e8\u5206\u533a\u57df, \u505a\u51cf\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u7ed3\u679c\u4e86, \u6574\u4e2a\u51cf\u6cd5\u5982\u4e0b:
\n$(a+c)*(b+d)-2*(1\/2ab)-2*(1\/2cd)-2cb=ad-bc$<\/p>\n
\u5bf9\u4e8e\u77e9\u9635 $A$ :
\n$$
\nA=\\left(\\begin{array}{ll}
\n2 & 1 \\\\
\n1 & 3
\n\\end{array}\\right)
\n$$<\/p>\n
\u8ba1\u7b97\u884c\u5217\u5f0f:
\n$$
\n\\operatorname{det}(A)=2 \\cdot 3-1 \\cdot 1=6-1=5
\n$$<\/p>\n
\u56e0\u6b64\uff0c\u77e9\u9635 $A$ \u6784\u6210\u7684\u5e73\u884c\u56db\u8fb9\u5f62\u7684\u9762\u79ef\u662f\u884c\u5217\u5f0f\u7684\u7edd\u5bf9\u503c\uff0c\u5373:
\n$$
\n|\\operatorname{det}(A)|=5
\n$$<\/p>\n
\u8fd9\u4e0e\u884c\u5217\u5f0f\u7684\u5b9a\u4e49\u76f8\u7b26\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 Python \u6765\u9a8c\u8bc1\u8fd9\u4e00\u8ba1\u7b97\uff1a<\/p>\n
import numpy as np\n\n# \u5b9a\u4e49\u77e9\u9635 A\nA = np.array([[2, 1], [1, 3]])\n\n# \u8ba1\u7b97\u884c\u5217\u5f0f\ndet_A = np.linalg.det(A)\nint(det_A)\n<\/code><\/pre>\n5<\/code><\/pre>\n\u5728\u4e0a\u9762\u7684\u57fa\u7840\u4e0a\u6765\u8003\u8651\u884c\u5217\u5f0f\u4e3a0\uff0c\u62161\u65f6\u5019\uff0c\u4ee3\u8868\u4ec0\u4e48?<\/strong><\/p>\n\u884c\u5217\u5f0f\u4e3a\u96f6\uff1a<\/p>\n
\u884c\u5217\u5f0f\u7684\u51e0\u4f55\u610f\u4e49\u662f\u7ebf\u6027\u53d8\u6362\u5bf9\u9762\u79ef\u6216\u4f53\u79ef\u7684\u7f29\u653e\u6bd4\u4f8b\u56e0\u5b50\u3002\u5f53\u884c\u5217\u5f0f\u4e3a\u96f6\u65f6\uff0c\u610f\u5473\u7740\u53d8\u6362\u540e\u7684\u9762\u79ef\u6216\u4f53\u79ef\u4e3a\u96f6\uff0c\u5373\u539f\u6765\u7684\u51e0\u4f55\u7ed3\u6784\u88ab\u5b8c\u5168\u538b\u7f29\uff0c\u6ca1\u6709\u6269\u5c55\u5230\u539f\u6765\u7684\u7ef4\u5ea6\u3002<\/p>\n
\u5047\u8bbe\u77e9\u9635 $A$ \u4e3a:
\n$$
\nA=\\left(\\begin{array}{ll}
\n1 & 2 \\\\
\n2 & 4
\n\\end{array}\\right)
\n$$<\/p>\n
\u8ba1\u7b97\u884c\u5217\u5f0f:
\n$$
\n\\operatorname{det}(A)=1 \\cdot 4-2 \\cdot 2=4-4=0
\n$$<\/p>\n
import matplotlib.pyplot as plt\nimport numpy as np\n\n# \u5b9a\u4e49\u539f\u59cb\u5355\u4f4d\u6b63\u65b9\u5f62\u7684\u9876\u70b9\nsquare = np.array([[0, 0], [1, 0], [1, 1], [0, 1], [0, 0]])\n\n# \u5b9a\u4e49\u884c\u5217\u5f0f\u4e3a\u96f6\u7684\u77e9\u9635 A\nA = np.array([[1, 2], [2, 4]])\n\n# \u7ebf\u6027\u53d8\u6362\u540e\u7684\u9876\u70b9\ntransformed_square = square @ A.T\n\n# \u7ed8\u5236\u539f\u59cb\u5355\u4f4d\u6b63\u65b9\u5f62\nplt.plot(square[:, 0], square[:, 1], 'b-', label='orginal')\n\n# \u7ed8\u5236\u7ebf\u6027\u53d8\u6362\u540e\u7684\u56fe\u5f62\nplt.plot(transformed_square[:, 0], transformed_square[:, 1], 'r-', label='transformed')\n\n# \u7ed8\u5236\u5750\u6807\u8f74\nplt.axhline(0, color='black', linewidth=0.5)\nplt.axvline(0, color='black', linewidth=0.5)\n\n# \u7ed8\u5236\u70b9\nplt.scatter([0, A[0, 0], A[0, 1]], [0, A[1, 0], A[1, 1]], color='r')\n\n# \u6807\u6ce8\u70b9\nplt.text(0, 0, 'O', fontsize=12, ha='right')\nplt.text(A[0, 0], A[1, 0], 'A', fontsize=12, ha='right')\nplt.text(A[0, 1], A[1, 1], 'B', fontsize=12, ha='right')\n\n# \u8bbe\u7f6e\u56fe\u5f62\u5c5e\u6027\nplt.grid(True)\nplt.legend() \nplt.xlabel('x')\nplt.ylabel('y')\n\n# \u663e\u793a\u56fe\u5f62\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_39_0.png\")
\n<\/p>\n
\u5f53\u4e00\u4e2a\u77e9\u9635\u7684\u884c\u5217\u5f0f\u4e3a1\u65f6\uff0c\u51e0\u4f55\u4e0a\u610f\u5473\u7740\u8be5\u77e9\u9635\u5bf9\u5e94\u7684\u7ebf\u6027\u53d8\u6362\u4fdd\u6301\u9762\u79ef\uff08\u6216\u4f53\u79ef\uff09\u4e0d\u53d8\uff0c\u540c\u65f6\u4e5f\u4fdd\u6301\u65b9\u5411\u4e0d\u53d8\u3002\u7b80\u5355\u6765\u8bf4\uff0c\u8fd9\u79cd\u53d8\u6362\u53ea\u5bf9\u7a7a\u95f4\u8fdb\u884c\u65cb\u8f6c\u6216\u62c9\u4f38\uff0c\u800c\u4e0d\u4f1a\u6539\u53d8\u9762\u79ef\u6216\u4f53\u79ef\u3002<\/p>\n
import matplotlib.pyplot as plt\nimport numpy as np\n\n# \u5b9a\u4e49\u539f\u59cb\u5355\u4f4d\u6b63\u65b9\u5f62\u7684\u9876\u70b9\nsquare = np.array([[0, 0], [1, 0], [1, 1], [0, 1], [0, 0]])\n\n# \u5b9a\u4e49\u884c\u5217\u5f0f\u4e3a1\u7684\u77e9\u9635 A\nA = np.array([[1, 2], [0, 1]])\n\n# \u7ebf\u6027\u53d8\u6362\u540e\u7684\u9876\u70b9\ntransformed_square = square @ A.T\n\n# \u7ed8\u5236\u539f\u59cb\u5355\u4f4d\u6b63\u65b9\u5f62\nplt.plot(square[:, 0], square[:, 1], 'b-', label='orginal')\n\n# \u7ed8\u5236\u7ebf\u6027\u53d8\u6362\u540e\u7684\u56fe\u5f62\nplt.plot(transformed_square[:, 0], transformed_square[:, 1], 'r-', label='transformed')\n\n# \u7ed8\u5236\u5750\u6807\u8f74\nplt.axhline(0, color='black', linewidth=0.5)\nplt.axvline(0, color='black', linewidth=0.5)\n\n# \u7ed8\u5236\u70b9\nplt.scatter([0, A[0, 0], A[0, 1]], [0, A[1, 0], A[1, 1]], color='r')\n\n# \u6807\u6ce8\u70b9\nplt.text(0, 0, 'O', fontsize=12, ha='right')\nplt.text(A[0, 0], A[1, 0], 'A', fontsize=12, ha='right')\nplt.text(A[0, 1], A[1, 1], 'B', fontsize=12, ha='right')\n\n# \u8bbe\u7f6e\u56fe\u5f62\u5c5e\u6027\nplt.grid(True)\nplt.legend()\nplt.xlabel('x')\nplt.ylabel('y')\n\n# \u663e\u793a\u56fe\u5f62\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_41_0.png\")
\n<\/p>\n
\u77e9\u9635\u7684\u79e9<\/strong><\/p>\n\u79e9\u8868\u793a\u4ec0\u4e48\u5462? \u5047\u8bbe\u56db\u4e2a\u884c\u5411\u91cf\u7ec4\u6210\u7684\u77e9\u9635 $\\boldsymbol{A}$ \u5982\u4e0b:
\n$$
\n\\begin{aligned}
\n\\boldsymbol{\\alpha}_1 & =(1,1,3,1), \\boldsymbol{\\alpha}_2=(0,2,-1,4), \\\\
\n\\boldsymbol{\\alpha}_3 & =(0,0,0,5), \\boldsymbol{\\alpha}_4=(0,0,0,0) . \\\\
\n\\boldsymbol{A} & =\\left[\\begin{array}{cccc}
\n1 & 1 & 3 & 1 \\\\
\n0 & 2 & -1 & 4 \\\\
\n0 & 0 & 0 & 5 \\\\
\n0 & 0 & 0 & 0
\n\\end{array}\\right]
\n\\end{aligned}
\n$$<\/p>\n
\u6c42\u5176\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\uff1a<\/p>\n
\u5047\u8bbe\u6709: $k_1 \\boldsymbol{\\alpha}_1+k_2 \\boldsymbol{\\alpha}_2+k_3 \\boldsymbol{\\alpha}_3=0$ \uff08\u56e0\u4e3a $\\boldsymbol{\\alpha}_4$ \u662f\u96f6\u5411\u91cf, \u8ddf\u8c01\u90fd\u6709\u5173, \u6240\u4ee5\u53ea\u5047\u8bbe\u524d\u4e09\u4e2a\u5411\u91cf\u7ebf\u6027\u76f8\u5173\uff09\u3002<\/p>\n
$$
\n\\begin{cases}
\nk_1=0, \\\\
\nk_1+2 k_2=0, \\\\
\n3 k_1-k_2=0, \\\\
\nk_1+4 k_2+5 k_3=0
\n\\end{cases}
\n$$
\n\u89e3\u5f97: $k_1=k_2=k_3=0$, \u5373 $\\alpha_1, \\alpha_2, \\alpha_3$ \u7ebf\u6027\u65e0\u5173\u3002<\/p>\n
\u77e9\u9635\u7684\u79e9\u8868\u793a\u5f53\u524d\u77e9\u9635\u4e2d\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\u7ec4\u7684\u4e2a\u6570, \u5728\u5f53\u524d\u4f8b\u5b50\u4e2d\u5373\u4e3a 3 \u3002<\/p>\n
\u5728\u4e4b\u524d\u8bf4\u8fc7\u77e9\u9635\u53ef\u4ee5\u770b\u4f5c\u5bf9\u5411\u91cf\u505a\u53d8\u6362, \u4f8b\u5982\u53ef\u4ee5\u5bf9\u4e8c\u7ef4\u56fe\u5f62\u8fdb\u884c\u65cb\u8f6c, \u6bd4\u5982\u7528\u65cb\u8f6c\u77e9\u9635 $\\left[\\begin{array}{cc}\\cos (\\theta) & -\\sin (\\theta) \\\\ \\sin (\\theta) & \\cos (\\theta)\\end{array}\\right]$ \u3002<\/p>\n
\u6b64\u65f6\u7684\u65cb\u8f6c\u77e9\u9635\u79e9\u4e3a 2, \u53d8\u6362\u540e\u7684\u6548\u679c\u5982\u4e0b\u3002<\/p>\n
import matplotlib.pyplot as plt\nimport numpy as np\n\n# \u5b9a\u4e49\u65cb\u8f6c\u77e9\u9635\ntheta = np.pi \/ 4 # 45 degrees\nrotation_matrix = np.array([\n [np.cos(theta), -np.sin(theta)],\n [np.sin(theta), np.cos(theta)]\n])\n\nA = np.array([[1, -1], [1, -1]])\n\n# \u5b9a\u4e49\u4e00\u4e2a\u4e8c\u7ef4\u56fe\u5f62\uff08\u4f8b\u5982\u4e00\u4e2a\u6b63\u65b9\u5f62\uff09\nsquare = np.array([[1, 1], [-1, 1], [-1, -1], [1, -1], [1, 1]])\n\n# \u65cb\u8f6c\u540e\u7684\u56fe\u5f62\nrotated_square = square @ rotation_matrix.T\n\n# \u7ed8\u5236\u539f\u59cb\u56fe\u5f62\nplt.plot(square[:, 0], square[:, 1], 'b-', label='orginal')\n\n# \u7ed8\u5236\u65cb\u8f6c\u540e\u7684\u56fe\u5f62\nplt.plot(rotated_square[:, 0], rotated_square[:, 1], 'r-', label='transformed')\n\n# \u8bbe\u7f6e\u56fe\u5f62\u5c5e\u6027\nplt.axhline(0, color='black', linewidth=0.5)\nplt.axvline(0, color='black', linewidth=0.5)\nplt.grid(True)\nplt.legend() \nplt.xlabel('x')\nplt.ylabel('y')\n\n# \u663e\u793a\u56fe\u5f62\nplt.show()\n<\/code><\/pre>\n\n ![](\"http:\/\/www.gnn.club\/wp-content\/uploads\/2024\/07\/output_43_0.png\")
\n<\/p>\n
\u53d8\u6362\u540e\u7684\u7ed3\u679c\u4f9d\u7136\u662f\u4e8c\u7ef4\u7684\u3002\u5982\u679c\u7528\u77e9\u9635 $\\left[\\begin{array}{ll}1 & -1 \\\\ 1 & -1\\end{array}\\right]$ \u8fdb\u884c\u53d8\u6362\u5462? \u6b64\u65f6\u77e9\u9635\u7684\u79e9\u4e3a 1 , \u53d8\u6362\u6548\u679c\u5982\u4f55\uff1f<\/p>\n
\u53d8\u6362\u540e\u7684\u7ed3\u679c\u53d8\u6210\u4e86\u4e00\u7ef4\u3002\u8fd9\u91cc\u5c31\u4f53\u73b0\u51fa\u77e9\u9635\u7684\u79e9\u5bf9\u5411\u91cf\u7684\u53d8\u6362\u4f5c\u7528\u3002\u5373\uff1a\u5982\u679c\u77e9\u9635\u7684\u79e9\u4f4e\u4e8e\u5411\u91cf\u7a7a\u95f4\u7684\u7ef4\u5ea6\uff0c\u90a3\u4e48\u4f1a\u5bf9\u5411\u91cf\u8fdb\u884c\u964d\u7ef4\u3002<\/p>\n
\u6700\u540e\u5f3a\u8c03\u4e00\u4e0b\uff0c\u77e9\u9635\u7684\u79e9\u5b9e\u9645\u4e0a\u4ee3\u8868\u4e86\u77e9\u9635\u4e2d\u4e0d\u91cd\u590d\u7684\u4e3b\u8981\u7279\u5f81\u4e2a\u6570\u3002<\/p>\n
\u4e3e\u4e2a\u751f\u6d3b\u4e2d\u7684\u4f8b\u5b50\uff1a\u5bb6\u91cc\u7684\u6709\u4e09\u53ea\u5c0f\u732b\u54aa\uff0c\u7ed9\u5b83\u4eec\u62cd\u6444\u4e86100\u5f20\u7167\u7247\uff0c\u7ec4\u6210\u4e86\u5341\u884c\u5341\u5217\u7684\u77e9\u9635\uff0c\u8be5\u77e9\u9635\u7684\u79e9\u7b49\u4e8e3\uff0c\u5c31\u7b97\u62cd\u4e00\u5343\u5f20\u7167\u7247\uff0c\u7ec4\u6210\u7684\u77e9\u9635\u79e9\u8fd8\u662f3\u3002<\/p>\n
\u77e9\u9635\u7684\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf<\/strong><\/p>\n\u901a\u4fd7\u7684\u7406\u89e3\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u63cf\u8ff0\u4e86\u4ec0\u4e48\uff0c\u600e\u4e48\u83b7\u5f97\u6210\u529f\u7684\u4eba\u751f\uff1f\u5728\u6b63\u786e\u7684\u9053\u8def\u4e0a\u575a\u6301\u52aa\u529b\u4e0b\u53bb\u3002\u53ef\u4ee5\u628a\u5343\u767e\u79cd\u4eba\u751f\u9009\u62e9\u770b\u4f5c\u7279\u5f81\u5411\u91cf\uff08\u5b83\u662f\u6709\u65b9\u5411\u7684\uff01\uff09\uff1b\u628a\u5728\u8fd9\u4e2a\u65b9\u5411\u4e0a\u7684\u52aa\u529b\u770b\u4f5c\u7279\u5f81\u503c\uff08\u5b83\u662f\u4e00\u4e2a\u8861\u91cf\u5927\u5c0f\u7684\u91cf\uff09\u3002<\/p>\n
\u4e0b\u9762\u5f15\u51fa\u5b83\u7684\u6570\u5b66\u5b9a\u4e49: \u5bf9\u4e8e\u7ed9\u5b9a\u77e9\u9635 $\\boldsymbol{A}$, \u5bfb\u627e\u4e00\u4e2a\u5e38\u6570 $\\lambda$ \u548c\u975e\u96f6\u5411\u91cf $\\boldsymbol{x}$, \u4f7f\u5f97\u5411\u91cf $\\boldsymbol{x}$\u88ab\u77e9\u9635 $\\boldsymbol{A}$ \u4f5c\u7528\u540e, \u6240\u5f97\u7684\u5411\u91cf $\\boldsymbol{A x}$ \u4e0e\u539f\u5411\u91cf $\\boldsymbol{x}$ \u5e73\u884c, <\/p>\n
\u5e76\u4e14\u6ee1\u8db3 $\\boldsymbol{A x}=\\lambda \\boldsymbol{x}$ \u3002\u5176\u4e2d, $\\boldsymbol{x}$ \u662f\u7279\u5f81\u5411\u91cf, $\\lambda$ \u662f\u7279\u5f81\u503c, \u7279\u5f81\u503c\u8d8a\u5927\u8868\u793a\u8be5\u7279\u5f81\u5411\u91cf\u8d8a\u91cd\u8981\u3002<\/p>\n
\u4e3e\u4e2a\u4f8b\u5b50\u6765\u7406\u89e3:<\/p>\n
\u5411\u91cf $\\boldsymbol{e}_1=\\left[\\begin{array}{l}1 \\\\ 0\\end{array}\\right]$ \u548c\u5411\u91cf $\\boldsymbol{e}_2=\\left[\\begin{array}{l}0 \\\\ 1\\end{array}\\right]$ \u90fd\u662f\u5411\u91cf $\\boldsymbol{A}=\\left[\\begin{array}{ll}3 & 0 \\\\ 0 & 2\\end{array}\\right]$ \u7684\u7279\u5f81\u5411\u91cf\u3002<\/p>\n
\u56e0\u4e3a\u5b83\u4eec\u90fd\u53ef\u5199\u6210 $\\boldsymbol{A} \\boldsymbol{x}=\\lambda \\boldsymbol{x}$ \u7684\u5f62\u5f0f:
\n$$
\n\\begin{aligned}
\nA e_1 & =\\left[\\begin{array}{ll}
\n3 & 0 \\\\
\n0 & 2
\n\\end{array}\\right]\\left[\\begin{array}{l}
\n1 \\\\
\n0
\n\\end{array}\\right]=\\left[\\begin{array}{l}
\n3 \\\\
\n0
\n\\end{array}\\right]=3 e_1 \\\\
\nA e_2 & =\\left[\\begin{array}{ll}
\n3 & 0 \\\\
\n0 & 2
\n\\end{array}\\right]\\left[\\begin{array}{l}
\n0 \\\\
\n1
\n\\end{array}\\right]=\\left[\\begin{array}{l}
\n0 \\\\
\n2
\n\\end{array}\\right]=2 e_2
\n\\end{aligned}
\n$$<\/p>\n
\u7279\u5f81\u5411\u91cf\u6709\u65e0\u6570\u4e2a\uff0c\u4e14\u6b64\u65f6\u7279\u5f81\u5411\u91cf\u5728\u5e94\u7528\u539f\u59cb\u5411\u91cfA\u540e\uff0c\u53ea\u6709\u4f38\u7f29\u4f5c\u7528\uff0c\u6ca1\u6709\u65cb\u8f6c\u4f5c\u7528\uff0c\u4f38\u7f29\u7684\u6bd4\u4f8b\u5c31\u662f\u7279\u5f81\u503c\u3002\u5177\u4f53\u6765\u8bf4\uff1a<\/p>\n
\n- \u5bf9\u4e8e $\\mathbf{e}_1$ \uff0c\u77e9\u9635 $A$ \u5c06\u5176\u62c9\u4f38 3 \u500d\uff08\u7279\u5f81\u503c\u4e3a 3 \uff09\u3002<\/li>\n
- \u5bf9\u4e8e $\\mathbf{e}_2$ \uff0c\u77e9\u9635 $A$ \u5c06\u5176\u62c9\u4f38 2 \u500d\uff08\u7279\u5f81\u503c\u4e3a 2 \uff09\u3002<\/li>\n<\/ul>\n
\u6700\u540e\uff0c\u600e\u4e48\u6c42\u89e3\u7279\u5f81\u5411\u91cf\u5462\uff1f\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n
$$
\n\\begin{aligned}
\n& A \\boldsymbol{x}=\\lambda \\boldsymbol{x} \\\\
\n& \\boldsymbol{A} \\boldsymbol{x}=\\lambda(\\boldsymbol{I} \\boldsymbol{x})=\\lambda \\boldsymbol{I} \\boldsymbol{x} \\\\
\n& (\\boldsymbol{A}-\\lambda \\boldsymbol{I}) \\boldsymbol{x}=\\mathbf{0}
\n\\end{aligned}
\n$$<\/p>\n
\u6b64\u65f6\uff0c\u89e3\u7279\u5f81\u5411\u91cf\u53d8\u6210\u4e86\u6c42\u89e3\u9f50\u6b21\u65b9\u7a0b\u7684\u95ee\u9898\u3002<\/p>\n
\u4f8b\u5982\u6c42 $\\boldsymbol{A}=\\left[\\begin{array}{ll}2 & 0 \\\\ 0 & 3\\end{array}\\right]$ \u7684\u7279\u5f81\u503c\u3002
\n$$
\n\\begin{aligned}
\n& |\\boldsymbol{A}-\\lambda \\boldsymbol{I}|=0 \\\\
\n& \\boldsymbol{X}=\\left[\\begin{array}{ll}
\n2 & 0 \\\\
\n0 & 3
\n\\end{array}\\right]-\\lambda \\boldsymbol{I}=\\left[\\begin{array}{cc}
\n2-\\lambda & 0 \\\\
\n0 & 3-\\lambda
\n\\end{array}\\right] \\\\
\n& \\operatorname{det}(\\boldsymbol{X})=(2-\\lambda) *(3-\\lambda)
\n\\end{aligned}
\n$$<\/p>\n
\u5f88\u5bb9\u6613\u770b\u51fa $\\lambda_1=2, \\lambda_2=3$ \u3002\u628a $\\lambda$ \u4ee3\u5165\u5f0f $\\boldsymbol{A} \\boldsymbol{v}=\\lambda \\boldsymbol{v}$ :<\/p>\n
$\\left[\\begin{array}{ll}2 & 0 \\\\ 0 & 3\\end{array}\\right]\\left[\\begin{array}{l}v_i \\\\ v_j\\end{array}\\right]=\\left[\\begin{array}{l}2 v_i \\\\ 3 v_j\\end{array}\\right]=2 \\cdot\\left[\\begin{array}{l}v_i \\\\ v_j\\end{array}\\right]$ \u6b64\u5f0f\u5728 $v_j=0$ \u65f6\u5bf9\u4efb\u4f55 $v_i$ \u6210\u7acb<\/p>\n
$\\left[\\begin{array}{ll}2 & 0 \\\\ 0 & 3\\end{array}\\right]\\left[\\begin{array}{l}v_i \\\\ v_j\\end{array}\\right]=\\left[\\begin{array}{l}2 v_i \\\\ 3 v_j\\end{array}\\right]=3 \\cdot\\left[\\begin{array}{l}v_i \\\\ v_j\\end{array}\\right]$ \u6b64\u5f0f\u5728 $v_i=0$ \u65f6\u5bf9\u4efb\u4f55 $v_j$ \u6210\u7acb<\/p>\n
\u8bf4\u660e, $x$ \u8f74\u548c $y$ \u8f74\u4e0a\u6240\u6709\u7684\u5411\u91cf\u90fd\u662f\u7279\u5f81\u5411\u91cf, \u4e14\u7ecf\u8fc7\u77e9\u9635 $\\boldsymbol{A}$ \u7684\u4f5c\u7528, \u4f1a\u5728 $x$ \u8f74\u4e0a\u62c9\u4f38\u4e24\u500d, \u5728 $y$ \u8f74\u4e0a\u62c9\u4f38 3 \u500d\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
Deep Learning Math \u7ebf\u6027\u4ee3\u6570\uff08Linear Algebra Tutorial\uff09 \u7ebf\u6027\u4ee3\u6570\u5728\u6df1 […]<\/p>\n","protected":false},"author":1,"featured_media":1637,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-1303","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-14"],"_links":{"self":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1303","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1303"}],"version-history":[{"count":150,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1303\/revisions"}],"predecessor-version":[{"id":1724,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1303\/revisions\/1724"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=\/wp\/v2\/media\/1637"}],"wp:attachment":[{"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1303"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1303"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}