{"id":313,"date":"2023-07-06T08:31:48","date_gmt":"2023-07-06T08:31:48","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e7%b2%97%e9%ba%bb%e5%b8%83%e7%9f%a9%e9%98%b5-%e7%b2%97%e9%ba%bb%e5%b8%83-2x2-3x3\/"},"modified":"2023-07-06T08:31:48","modified_gmt":"2023-07-06T08:31:48","slug":"%e7%b2%97%e9%ba%bb%e5%b8%83%e7%9f%a9%e9%98%b5-%e7%b2%97%e9%ba%bb%e5%b8%83-2x2-3x3","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e7%b2%97%e9%ba%bb%e5%b8%83%e7%9f%a9%e9%98%b5-%e7%b2%97%e9%ba%bb%e5%b8%83-2x2-3x3\/","title":{"rendered":"Hessian \u77e9\u9635\uff08\u6216 hesse\uff09"},"content":{"rendered":"<p>\u672c\u9875\u65e0\u7591\u662f\u73b0\u6709\u7684 Hessian \u77e9\u9635\u6700\u5b8c\u6574\u7684\u89e3\u91ca\u3002\u8fd9\u91cc\u89e3\u91ca\u4e86Hessian\u77e9\u9635\u7684\u6982\u5ff5\uff0c\u5982\u4f55\u7528\u4f8b\u5b50\u8ba1\u7b97\u5b83\uff0c\u751a\u81f3\u8fd8\u6709\u51e0\u4e2a\u5df2\u89e3\u51b3\u7684\u7ec3\u4e60\u53ef\u4f9b\u7ec3\u4e60\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u4e86\u89e3\u5982\u4f55\u8ba1\u7b97\u591a\u53d8\u91cf\u51fd\u6570\u7684\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c\uff0c\u4ee5\u53ca\u5b83\u662f\u51f9\u51fd\u6570\u8fd8\u662f\u51f8\u51fd\u6570\u3002\u6700\u540e\uff0c\u60a8\u8fd8\u5c06\u627e\u5230 Hessian \u77e9\u9635\u5b9e\u7528\u7a0b\u5e8f\u548c\u5e94\u7528\u7a0b\u5e8f\u3002<\/p>\n<h2 class=\"wp-block-heading\">\u4ec0\u4e48\u662f Hessian \u77e9\u9635\uff1f<\/h2>\n<p> Hessian\uff08\u6216Hessian\uff09\u77e9\u9635\u7684\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> <strong>Hessian \u77e9\u9635<\/strong>\u662f\u7531 n \u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u7684\u504f\u4e8c\u9636\u5bfc\u6570\u7ec4\u6210\u7684\u7ef4\u5ea6\u4e3a n \u00d7 n \u7684\u65b9\u9635\u3002<\/p>\n<p>\u8fd9\u4e2a\u77e9\u9635\u4e5f\u88ab\u79f0\u4e3aHessian\uff0c\u751a\u81f3\u5728\u4e00\u4e9b\u6570\u5b66\u4e66\u7c4d\u4e2d\u5b83\u88ab\u547d\u540d\u4e3aDiscriminant\u3002\u4f46\u6700\u5e38\u89c1\u7684\u79f0\u547c\u65b9\u5f0f\u662f Hessian \u77e9\u9635\u3002<\/p>\n<p>\u56e0\u6b64Hessian\u77e9\u9635\u7684\u516c\u5f0f\u5982\u4e0b\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-matricielle-hessienne.webp\" alt=\"Hessian \u6216 Hessian \u77e9\u9635\u516c\u5f0f\" class=\"wp-image-2445\" width=\"541\" height=\"345\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u56e0\u6b64\uff0cHessian \u77e9\u9635\u59cb\u7ec8\u662f\u65b9\u9635\uff0c\u5176\u7ef4\u5ea6\u7b49\u4e8e\u51fd\u6570\u4e2d\u53d8\u91cf\u7684\u6570\u91cf\u3002\u4f8b\u5982\uff0c\u5982\u679c\u51fd\u6570\u6709 3 \u4e2a\u53d8\u91cf\uff0c\u5219 Hessian \u77e9\u9635\u7684\u7ef4\u5ea6\u4e3a 3\u00d73\u3002<\/p>\n<p>\u6b64\u5916\uff0c<strong>\u65bd\u74e6\u8328\u5b9a\u7406<\/strong>\uff08\u6216\u514b\u83b1\u7f57\u5b9a\u7406\uff09\u6307\u51fa\uff0c\u5fae\u5206\u7684\u987a\u5e8f\u5e76\u4e0d\u91cd\u8981\uff0c\u5373\u9996\u5148\u5bf9\u53d8\u91cf\u8fdb\u884c\u90e8\u5206\u63a8\u5bfc<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01a7b7b5dca66cb33a1207e1f39c1140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p>\u90a3\u4e48\u5173\u4e8e\u53d8\u91cf<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1cd6be340b4fce14489cf5b565a169e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p>\u76f8\u5f53\u4e8e\u90e8\u5206\u5fae\u5206<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1cd6be340b4fce14489cf5b565a169e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p>\u7136\u540e\u5c0a\u91cd<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01a7b7b5dca66cb33a1207e1f39c1140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p>\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e5cc28564c40c6588680df48d8255ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x_i\\partial x_j} = \\cfrac{\\partial^2 f}{\\partial x_j\\partial x_i}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"133\" style=\"vertical-align: -18px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0cHessian\u77e9\u9635\u662f\u4e00\u4e2a<strong>\u5bf9\u79f0<\/strong>\u77e9\u9635\uff0c\u6216\u8005\u6362\u53e5\u8bdd\u8bf4\uff0c\u5b83\u5177\u6709\u4ee5\u5176\u4e3b\u5bf9\u89d2\u7ebf\u4e3a\u8f74\u7684\u5bf9\u79f0\u6027\u3002<\/p>\n<p>\u51fa\u4e8e\u597d\u5947\uff0cHessian \u77e9\u9635\u662f\u4ee5 19 \u4e16\u7eaa\u5fb7\u56fd\u6570\u5b66\u5bb6 Ludwig Otto Hesse \u7684\u540d\u5b57\u547d\u540d\u7684\uff0c\u4ed6\u5bf9\u7ebf\u6027\u4ee3\u6570\u9886\u57df\u505a\u51fa\u4e86\u975e\u5e38\u91cd\u8981\u7684\u8d21\u732e\u3002<\/p>\n<h2 class=\"wp-block-heading\">\u8ba1\u7b97Hessian\u77e9\u9635\u7684\u793a\u4f8b<\/h2>\n<p>\u8ba9\u6211\u4eec\u770b\u4e00\u4e2a\u5982\u4f55\u67e5\u627e 2 \u00d7 2 \u7ef4 Hessian \u77e9\u9635\u7684\u793a\u4f8b\uff1a<\/p>\n<ul>\n<li>\u8ba1\u7b97\u4ee5\u4e0b\u51fd\u6570\u7684 (1,0) \u70b9\u5904\u7684 Hessian \u77e9\u9635\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03bcc33e393ddc95defdc3cc04da35c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)=y^4+x^3+3x^2+ 4y^2 -4xy -5y +8\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"348\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u4e00\u9636\u504f\u5bfc\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fd57722dfa592f129cac21ba8f183e05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = 3x^2 +6x -4y\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"152\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1847d0b9ecd032b396d64ac27e5e3eeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = 4y^3+8y -4x -5\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"181\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e00\u9636\u5bfc\u6570\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8ba1\u7b97\u6240\u6709\u4e8c\u9636\u504f\u5bfc\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-294ce075a54a240e012d30770fa10e3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} = 6x +6\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"102\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-895c605e3c401f96a0a9b07ea105522c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =12y^2 +8\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"118\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2ea4b2979928198e9fb91592d9e38874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y} = \\cfrac{\\partial^2 f}{\\partial y \\partial x}= -4\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"153\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u4ece 2 \u00d7 2 \u77e9\u9635\u7684\u516c\u5f0f\u4e2d\u627e\u5230 Hessian \u77e9\u9635\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-926f350fe0ac3184ec0b563b57fd6041_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (x,y)=\\begin{pmatrix}\\cfrac{\\partial^2 f}{\\partial x^2} &amp; \\cfrac{\\partial^2 f}{\\partial x \\partial y} \\\\[4ex] \\cfrac{\\partial^2 f}{\\partial y \\partial x} &amp; \\cfrac{\\partial^2 f}{\\partial y^2} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"111\" width=\"215\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b7f3d45918645a5b6019896ed45eda75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (x,y)=\\begin{pmatrix}6x +6 &amp;-4 \\\\[2ex] -4 &amp; 12y^2+8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"246\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u5728 (1,0) \u70b9\u8ba1\u7b97\u7684 Hessian \u77e9\u9635\u5c06\u662f\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bdccfc61f7befe6c75f66c8a4658f3e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,0)=\\begin{pmatrix}6(1) +6 &amp;-4 \\\\[2ex] -4 &amp; 12(0)^2+8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"270\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-matrice-hessienne-de-dimension-22152.webp\" alt=\"\u7c97\u9ebb\u5e03\u6216\u7c97\u9ebb\u5e03\u6a21\u5177\u793a\u4f8b\" class=\"wp-image-2487\" width=\"230\" height=\"80\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\">\u89e3\u51b3\u4e86 Hessian \u77e9\u9635\u7684\u95ee\u9898<\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u51fd\u6570\u5728\u70b9 (1,1) \u5904\u5177\u6709 2 \u4e2a\u53d8\u91cf\u7684 Hessian \u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1e63eb1506d471fb8fdbc0e6db8de0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)=x^2y+y^2x\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"151\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u6c42\u51fd\u6570\u7684\u4e00\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9786ccaea4ad9305eb8b50d98f7f2626_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = 2xy+y^2\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"111\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b68a8a666732d228ad06ebd8bade063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = x^2+2yx\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"112\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u8ba1\u7b97\u4e86\u4e00\u9636\u5bfc\u6570\uff0c\u6211\u4eec\u5c31\u7ee7\u7eed\u6c42\u89e3\u6240\u6709\u4e8c\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef3e5008f32d204cae79f8faceafb6ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} = 2y\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"70\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e3f3f68ddab03ad22776b84f441cb7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =2x\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"71\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6383bbe85e851dd1e573981b52a0158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y} = \\cfrac{\\partial^2 f}{\\partial y \\partial x}=2x+2y\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"188\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64Hessian\u77e9\u9635\u5b9a\u4e49\u5982\u4e0b\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-matrice-hessian-22152-1.webp\" alt=\"\u7ef4\u5ea6\u4e3a 2x2 \u7684 Hessian \u6216 Hessian \u77e9\u9635\u7684\u6c42\u89e3\u7ec3\u4e60\" class=\"wp-image-2492\" width=\"290\" height=\"81\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u5269\u4e0b\u7684\u5c31\u662f\u8ba1\u7b97\u70b9 (1,1) \u5904\u7684 Hessian \u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5353c0229942269e07455047284f92b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix}2\\cdot 1 &amp;2 \\cdot 1+2\\cdot 1 \\\\[1.5ex] 2\\cdot 1+2\\cdot 1 &amp; 2\\cdot 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"292\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf00fccdb37a19388e76b5a84a408d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{H_f (1,1)}=\\begin{pmatrix}\\bm{2} &amp; \\bm{4} \\\\[1.1ex] \\bm{4} &amp; \\bm{2} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e602<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u51fd\u6570\u5728\u4e24\u4e2a\u53d8\u91cf\u4e2d\u7684\u70b9 (1,1) \u5904\u7684 Hessian\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d577030e612ef7b6ab7b59aea4469539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)= e^{y\\ln x}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"115\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u51fd\u6570\u7684\u4e00\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1025a749344fdf59c8c016387b7e2c37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = e^{y\\ln x} \\cdot \\cfrac{y}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"110\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2ddfad6313a350d43404b19a2278d771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = e^{y\\ln x} \\cdot \\ln x\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u6709\u4e86\u4e00\u9636\u5bfc\u6570\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8ba1\u7b97\u51fd\u6570\u7684\u4e8c\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-edcdf882fac053dc938c4f0d10060d30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} = e^{y\\ln x} \\cdot \\cfrac{y^2}{x^2} - e^{y\\ln x} \\cdot \\cfrac{y}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"219\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f59a0f7a852311293edd2231235ae7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =e^{y\\ln x} \\cdot \\ln ^2 x\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"141\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b760e29d744c2f294cf9cec33d61d1e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y}=\\cfrac{\\partial^2 f}{\\partial y \\partial x} =e^{y\\ln x} \\cdot \\cfrac{y}{x}\\cdot \\ln x + e^{y\\ln x}\\cdot \\cfrac{1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"323\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\u51fd\u6570\u7684 Hessian \u77e9\u9635\u662f\u7ef4\u5ea6\u4e3a 2\u00d72 \u7684\u65b9\u9635\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-hesse-de-dimension-22152-1.webp\" alt=\"\u5df2\u6c42\u89e3\u7684 Hessian \u7ec3\u4e60\u6216\u7ef4\u5ea6\u4e3a 2x2 \u7684 Hessian \u77e9\u9635\" class=\"wp-image-2516\" width=\"620\" height=\"135\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u5269\u4e0b\u7684\u5c31\u662f\u8ba1\u7b97\u70b9 (1,1) \u5904\u7684 Hessian \u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c316cc61e6d007e5d034274e0f494520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix} e^{1\\ln (1)} \\displaystyle \\cdot \\cfrac{1^2}{1^2} - e^{1\\ln (1)} \\cdot \\cfrac{1}{1^2}&amp; e^{1\\ln (1)} \\cdot \\cfrac{1}{1}\\cdot \\ln (1) + e^{1\\ln (1)}\\cdot \\cfrac{1}{1} \\\\[3ex] e^{1\\ln (1)} \\cdot \\cfrac{1}{1}\\cdot \\ln (1) + e^{1\\ln (1)}\\cdot \\cfrac{1}{1} &amp; e^{1\\ln (1)} \\cdot \\ln ^2 (1) \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"99\" width=\"557\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e456b856c722a140d73ade63f13ec9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix}e^{0} \\cdot 1 - e^{0} \\cdot 1&amp; e^{0} \\cdot 1\\cdot 0 + e^{0}\\cdot 1 \\\\[2ex] e^{0} \\cdot 1\\cdot 0 + e^{0}\\cdot 1 &amp; e^{0} \\cdot 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"366\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-929fbf6e7f0f90110d11d4ccd51fd51a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix}1 - 1&amp; 0+ 1 \\\\[1.5ex] 0 +1 &amp; 1 \\cdot 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"206\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce780ddb8c09515afccfb2da2d842584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{H_f (1,1)}=\\begin{pmatrix}\\bm{0} &amp; \\bm{1} \\\\[1.1ex] \\bm{1} &amp; \\bm{0} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e603<\/h3>\n<p>\u6c42\u8be5\u70b9\u7684 Hessian \u77e9\u9635<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b324eec3855aa704cfe7cef3a72713f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,1,\\pi)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u4ee5\u4e0b\u5177\u6709 3 \u4e2a\u53d8\u91cf\u7684\u51fd\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-07614221c6aa91fe6591ecfdd7ee064b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y,z)= e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"189\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u8ba1\u7b97\u51fd\u6570\u7684\u4e00\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3403bbb8643fb0c3f01dc93031683d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = -e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"155\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e77e08ad03b7a454f9a9f5c320d4902_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = ze^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"149\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-912b948c3339efba02beb10da8853e89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial z} = ye^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"149\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u6709\u4e86\u4e00\u9636\u5bfc\u6570\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8ba1\u7b97\u51fd\u6570\u7684\u4e8c\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-831ffaaf0208ac80132c58e97c10696d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} =e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"149\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f8c77e5f03861145b5370a208642c199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y}=\\cfrac{\\partial^2 f}{\\partial y \\partial x} =-ze^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"248\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3a5180d638ffbedb5028da56c5c2b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial z}=\\cfrac{\\partial^2 f}{\\partial z \\partial x} =-ye^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"247\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-001a3978b472e9085c4658d19e976079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =-z^2e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"179\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8c2c222f469c688e41ed773c1721834_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y \\partial z}=\\cfrac{\\partial^2 f}{\\partial z \\partial y} =e^{-x}\\cdot \\text{cos}(yz)-yze^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"360\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-98dcbe5198ef8336355c89a80bb981ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial^2 z} = -y^2e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"179\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5219\u51fd\u6570\u7684 Hessian \u77e9\u9635\u662f\u7ef4\u5ea6\u4e3a 3\u00d73 \u7684\u65b9\u9635\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-matrice-de-hesse-en-3-dimensions-32153-1.webp\" alt=\"\u5c3a\u5bf8\u4e3a 3x3 \u7684\u7c97\u9ebb\u5e03\u6216\u7c97\u9ebb\u5e03\u77e9\u9635\u7684\u793a\u4f8b\" class=\"wp-image-2537\" width=\"857\" height=\"109\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u5c06\u53d8\u91cf\u66ff\u6362\u4e3a\u5b83\u4eec\u5728\u8be5\u70b9\u5404\u81ea\u7684\u503c<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf0e780dfde4cefb749a03fe14266290_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,1,\\pi):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e198192f67babd81228caa53b66e8a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f(0,1,\\pi)=\\begin{pmatrix}e^{-0}\\cdot \\text{sen}(1\\pi) &amp; -\\pi e^{-0}\\cdot \\text{cos}(1\\pi) &amp;-1e^{-0}\\cdot \\text{cos}(1\\pi) \\\\[1.5ex] -\\pi e^{-0}\\cdot \\text{cos}(1 \\pi)&amp;-\\pi^2e^{-0}\\cdot \\text{sen}(1 \\pi) &amp;e^{-0}\\cdot \\text{cos}(1 \\pi)-1 \\pi e^{-0}\\cdot \\text{sen}(1 \\pi) \\\\[1.5ex] -1e^{-0}\\cdot \\text{cos}(1 \\pi)&amp; e^{-0}\\cdot \\text{cos}(1 \\pi)-1 \\pi e^{-0}\\cdot \\text{sen}(1 \\pi)&amp; -1^2e^{-0}\\cdot \\text{sen}(1 \\pi) \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"756\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4ce9c6b4cfcddfb0c2eb51db1189c653_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f(0,1,\\pi)=\\begin{pmatrix}1\\cdot 0 &amp; -\\pi \\cdot 1 \\cdot (-1)&amp;-1\\cdot 1 \\cdot (-1) \\\\[1.5ex] -\\pi \\cdot 1 \\cdot (-1) &amp;-\\pi^2\\cdot 1\\cdot 0 &amp;1 \\cdot (-1)-\\pi \\cdot 1\\cdot 0 \\\\[1.5ex] -1\\cdot 1 \\cdot (-1) &amp; 1\\cdot (-1) - \\pi \\cdot 1\\cdot 0 &amp; -1\\cdot 1 \\cdot 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"527\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-matrice-hessiana-32153-1.webp\" alt=\"\u7ec3\u4e60\u9010\u6b65\u89e3\u51b3\u5c3a\u5bf8\u4e3a 3x3 \u7684 Hessian \u6216 Hessian \u77e9\u9635\" class=\"wp-image-2536\" width=\"291\" height=\"109\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e604<\/h3>\n<p>\u786e\u5b9a\u4ee5\u4e0b\u5177\u6709 4 \u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u5728\u70b9 (2,-1,1,-1) \u5904\u7684 Hessian \u77e9\u9635\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4cf4dde4cb282d4f494142475a514b6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y,z,w)= 2x^3y^4zw^2 - 2y^3w^4+ 3x^2z^2\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"320\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u7b2c\u4e00\u6b65\u662f\u6c42\u51fd\u6570\u7684\u4e00\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d559c24695adc4e5bf25a07162b0b82c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} =6x^2y^4zw^2 + 6xz^2\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"175\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7e4b630227d3c64561c2a87f729612b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} =8x^3y^3zw^2 - 6y^2w^4\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"186\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c473febcf510cf11a4b30fad46fb3d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial z} = 2x^3y^4w^2 + 6x^2z\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"166\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-afc9ac6f800c06919a5bee6d6ea20a38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial w} =4x^3y^4zw - 8y^3w^3\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"181\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u6c42\u89e3\u51fd\u6570\u7684\u4e8c\u9636\u504f\u5bfc\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-55333bf53a5934f849b6eea4d5a4f64c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} =12xy^4zw^2 + 6z^2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"174\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ecd6fb72d35cb8c8b09e14dd2337bdfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y}=\\cfrac{\\partial^2 f}{\\partial y \\partial x}=24x^2y^3zw^2\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"211\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02885e698f3ae93bbf9515970de030ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial z}=\\cfrac{\\partial^2 f}{\\partial z \\partial x}=6x^2y^4w^2 + 12xz\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"252\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0a43e7868272e5f9981fd6961c4d2078_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial w} = \\cfrac{\\partial^2 f}{\\partial w \\partial x}=12x^2y^4zw\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"212\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fd1617aa3759b2cab9621b821d06f42d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =24x^3y^2zw^2 - 12yw^4\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"204\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c019a5c8dd396e6d0ee491d1e96b0f42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y \\partial z}=\\cfrac{\\partial^2 f}{\\partial y \\partial z}=8x^3y^3w^2\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"191\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-83dc31061c77785623e54e55f39c7bea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y \\partial w} = \\cfrac{\\partial^2 f}{\\partial w \\partial y}=16x^3y^3zw - 24y^2w^3\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"287\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-57b2a8d35948c0ac623709a00922f2a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial^2 z} =6x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"78\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-260eacc428eb6db6a9a5ad0b7c7e9649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial z \\partial w} = \\cfrac{\\partial^2 f}{\\partial w \\partial z}=4x^3y^4w\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"192\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d4cb2a8938dfd2d17cc68cd9260eae5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial^2 w} =4x^3y^4z - 24y^3w^2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"184\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\u6c42\u89e3\u6240\u6709\u504f\u5bfc\u6570\u5f97\u5230\u76844\u00d74 Hessian\u77e9\u9635\u7684\u8868\u8fbe\u5f0f\u4e3a\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-toile-de-jute-ou-matrice-de-toile-de-jute-de-dimension-44.webp\" alt=\"\u9010\u6b65\u6c42\u89e3\u5c3a\u5bf8\u4e3a 4x4 \u7684\u7c97\u9ebb\u5e03\u6216\u7c97\u9ebb\u5e03\u77e9\u9635\u7684\u793a\u4f8b\" width=\"854\" height=\"180\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u5c06\u672a\u77e5\u6570\u66ff\u6362\u4e3a\u5b83\u4eec\u5404\u81ea\u7684\u70b9\u503c\uff082\uff0c-1\uff0c1\uff0c-1\uff09\u5e76\u8fdb\u884c\u8ba1\u7b97\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-hesse-ou-matrice-hessienne-de-dimension-44.webp\" alt=\"\u9010\u6b65\u6c42\u89e3 4x4 \u7ef4 Hessian \u6216 Hessian \u77e9\u9635\u7684\u7ec3\u4e60\" width=\"457\" height=\"182\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h2 class=\"wp-block-heading\">\u5982\u4f55\u77e5\u9053Hessian\u77e9\u9635\u4f55\u65f6\u4e3a\u6b63\u3001\u8d1f\u6216\u4e0d\u5b9a\uff1f<\/h2>\n<p>\u6b63\u5982\u6211\u4eec\u7a0d\u540e\u5c06\u770b\u5230\u7684\uff0c\u77e5\u9053 Hessian \u77e9\u9635\u662f\u6b63\u534a\u5b9a\u3001\u6b63\u5b9a\u3001\u8d1f\u534a\u5b9a\u3001\u8d1f\u5b9a\u8fd8\u662f\u4e0d\u5b9a\u77e9\u9635\u662f\u975e\u5e38\u6709\u7528\u7684\u3002\u90a3\u4e48\u8ba9\u6211\u4eec\u770b\u770b\u5982\u4f55\u627e\u51fa\u7b54\u6848\uff1a<\/p>\n<h3 class=\"wp-block-heading\">\u7279\u5f81\u503c\uff08\u6216\u7279\u5f81\u503c\uff09\u7684\u51c6\u5219<\/h3>\n<p>\u77e5\u9053\u5b83\u662f\u4ec0\u4e48\u7c7b\u578b\u7684\u77e9\u9635\u7684\u4e00\u79cd\u65b9\u6cd5\u662f\u901a\u8fc7Hessian\u77e9\u9635\u7684\u7279\u5f81\u503c\uff08\u6216\u7279\u5f81\u503c\uff09\uff1a<\/p>\n<ul>\n<li>\u5982\u679c Hessian \u77e9\u9635\u7684\u7279\u5f81\u503c\uff08\u6216\u591a\u4e2a\u7279\u5f81\u503c\uff09\u7b49\u4e8e\u4e14\u5927\u4e8e\u96f6\uff0c\u5219\u8be5\u77e9\u9635\u662f<span style=\"color:#1976d2;\"><strong>\u6b63\u534a\u5b9a\u7684<\/strong><\/span>\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5b83\u5177\u6709\u6b63\u7279\u5f81\u503c\u4e14\u81f3\u5c11\u6709\u4e00\u4e2a\u7b49\u4e8e0\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-21d0afe89f5d1545ccda3d2bd0d8660a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda \\geq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<ul>\n<li>\u5982\u679c Hessian \u77e9\u9635\u7684\u6240\u6709\u7279\u5f81\u503c\uff08\u6216\u591a\u4e2a\u7279\u5f81\u503c\uff09\u90fd\u5b8c\u5168\u5927\u4e8e 0\uff08\u6b63\uff09\uff0c\u5219 Hessian \u77e9\u9635\u662f<span style=\"color:#1976d2;\"><strong>\u6b63\u5b9a\u7684<\/strong><\/span>\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f203da6c7fff0b1487c2084e3d90966b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda > 0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;14&#8243; width=&#8221;43&#8243; style=&#8221;vertical-align: -2px;&#8221;><\/p>\n<\/p>\n<ul>\n<li>\u5982\u679c Hessian \u77e9\u9635\u7684\u7279\u5f81\u503c\uff08\u6216\u591a\u4e2a\u7279\u5f81\u503c\uff09\u7b49\u4e8e\u4e14\u5c0f\u4e8e\u96f6\uff0c\u5219\u8be5<span style=\"color:#1976d2;\"><strong>\u77e9\u9635\u662f\u8d1f\u534a\u5b9a\u7684<\/strong><\/span>\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5b83\u5177\u6709\u8d1f\u7279\u5f81\u503c\u4e14\u81f3\u5c11\u6709\u4e00\u4e2a\u7b49\u4e8e0\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dfdf139bb553d680b38761b565dd3db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda \\leq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<ul>\n<li>\u5982\u679c Hessian \u77e9\u9635\u7684\u6240\u6709\u7279\u5f81\u503c\uff08\u6216\u591a\u4e2a\u7279\u5f81\u503c\uff09\u90fd\u5c0f\u4e8e 0\uff08\u8d1f\uff09\uff0c\u5219 Hessian \u77e9\u9635\u662f<span style=\"color:#1976d2;\"><strong>\u8d1f\u5b9a\u7684<\/strong><\/span>\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1fdc8fd3de7c5d7643ca8e4dbfe3704d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda < 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"43\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<ul>\n<li>\u5f53Hessian\u77e9\u9635\u6709\u6b63\u8d1f\u7279\u5f81\u503c\uff08\u6216\u7279\u5f81\u503c\uff09\u65f6\uff0c\u5b83\u662f<span style=\"color:#1976d2;\"><strong>\u672a\u5b9a\u4e49\u7684<\/strong><\/span>\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fa9eea2abe5f790e65f0c5afab8c3adb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda > 0 \\qquad \\lambda <0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"121\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\">\u897f\u5c14\u7ef4\u65af\u7279\u51c6\u5219<\/h3>\n<p>\u77e5\u9053Hessian\u77e9\u9635\u662f\u4ec0\u4e48\u7c7b\u578b\u7684\u53e6\u4e00\u79cd\u65b9\u6cd5\u662f\u4f7f\u7528\u897f\u5c14\u7ef4\u65af\u7279\u51c6\u5219\uff0c\u5c3d\u7ba1\u8fd9\u4e2a\u5b9a\u7406\u53ea\u80fd\u8ba9\u6211\u4eec\u77e5\u9053\u5b83\u662f\u6b63\u5b9a\u3001\u8d1f\u5b9a\u8fd8\u662f\u4e0d\u5b9a\u3002\u4f46\u6709\u65f6\u5b83\u7684\u4f7f\u7528\u901f\u5ea6\u4f1a\u66f4\u5feb\uff0c\u56e0\u4e3a\u8ba1\u7b97\u901a\u5e38\u66f4\u5bb9\u6613\u3002 <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<p>\u56e0\u6b64\uff0c<strong>\u897f\u5c14\u7ef4\u65af\u7279\u51c6\u5219<\/strong>\u5982\u4e0b\uff1a<\/p>\n<ul>\n<li>\u5982\u679cHessian\u77e9\u9635\u7684\u6240\u6709\u4e3b\u6b21\u6570\u90fd\u5927\u4e8e0\uff0c\u5219\u5b83\u662f<span style=\"color:#1976d2;\"><strong>\u6b63\u5b9a<\/strong><\/span>\u77e9\u9635\u3002<\/li>\n<\/ul>\n<ul>\n<li>\u5982\u679c Hessian \u77e9\u9635\u7684\u4e3b\u6b21\u6570\u4e3a\u5076\u6570\u4e14\u5927\u4e8e 0\uff0c\u4e14\u5947\u6570\u6570\u5c0f\u4e8e 0\uff0c\u5219\u4e3a<span style=\"color:#1976d2;\"><strong>\u8d1f\u5b9a<\/strong><\/span>\u77e9\u9635\u3002<\/li>\n<\/ul>\n<ul>\n<li>\u5982\u679cHessian\u77e9\u9635\u7684\u6240\u6709\u4e3b\u6b21\u6570\u90fd\u4e0d\u4e3a0\u5e76\u4e14\u524d\u4e24\u4e2a\u6761\u4ef6\u90fd\u4e0d\u6ee1\u8db3\uff0c\u5219\u5b83\u662f<span style=\"color:#1976d2;\"><strong>\u4e0d\u5b9a<\/strong><\/span>\u77e9\u9635\u3002 <\/li>\n<\/ul>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/majeurs-mineurs-d-une-matrice.webp\" alt=\"Hessian \u77e9\u9635\u7684\u4e3b\u6b21\u5b50\" class=\"wp-image-2583\" width=\"417\" height=\"361\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u663e\u7136\uff0cHessian\u77e9\u9635\u7684\u6700\u5927\u4e3b\u6b21\u6570\u603b\u662f\u4e0e\u5176\u884c\u5217\u5f0f\u4e00\u81f4\u3002\u4ec5\u4f9b\u53c2\u8003\uff0cHessian \u77e9\u9635\u7684\u884c\u5217\u5f0f\u4e5f\u79f0\u4e3a\u201cHessian\u201d\uff0c\u5c3d\u7ba1\u4e3a\u4e86\u907f\u514d\u6df7\u6dc6\u6211\u4eec\u5728\u8fd9\u91cc\u4e0d\u4f1a\u8fd9\u6837\u505a\u3002<\/p>\n<h2 class=\"wp-block-heading\">\u5982\u4f55\u4f7f\u7528 Hessian \u77e9\u9635\u8ba1\u7b97\u51fd\u6570\u7684\u6700\u5927\u503c\u6216\u6700\u5c0f\u503c<\/h2>\n<p>\u4e00\u65e6\u60a8\u77e5\u9053\u5982\u4f55\u8ba1\u7b97 Hessian \u77e9\u9635\uff0c\u60a8\u53ef\u80fd\u60f3\u77e5\u9053\uff1a\u8fd9\u4e2a\u77e9\u9635\u6709\u4ec0\u4e48\u7528\uff1f<\/p>\n<p>\u55ef\uff0cHessian \u77e9\u9635\u7684\u5e94\u7528\u4e4b\u4e00\u5c31\u662f\u6c42\u5177\u6709\u591a\u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u7684\u6700\u5927\u503c\u6216\u6700\u5c0f\u503c\u3002\u4ee5\u4e0b\u662f\u5982\u4f55\u8ba1\u7b97\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c\u7684\u5206\u6b65\u8bf4\u660e\uff1a<\/p>\n<ol style=\"color:#1976d2; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">\u9996\u5148\uff0c\u8ba1\u7b97\u591a\u53d8\u91cf\u51fd\u6570\u7684<strong>\u4e34\u754c\u70b9<\/strong>\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u8ba1\u7b97\u51fd\u6570\u7684\u68af\u5ea6\u6216<a href=\"https:\/\/mathority.org\/cn\/\u96c5\u53ef\u6bd4\u77e9\u9635-\u96c5\u53ef\u6bd4\u77e9\u9635\/\">\u96c5\u53ef\u6bd4\u77e9\u9635<\/a>\uff0c\u5c06\u5176\u8bbe\u7f6e\u4e3a 0 \u5e76\u6c42\u89e3\u65b9\u7a0b\u3002<\/span><\/li>\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">\u8ba1\u7b97Hessian\u77e9\u9635\u3002<\/span><\/li>\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">\u5c06\u6b65\u9aa41\u4e2d\u627e\u5230\u7684\u4e34\u754c\u70b9\u4ee3\u5165Hessian\u77e9\u9635\u4e2d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u83b7\u5f97\u4e0e\u4e34\u754c\u70b9\u5177\u6709\u8be5\u51fd\u6570\u4e00\u6837\u591a\u7684 Hessian \u77e9\u9635\u3002<\/span><\/li>\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">\u6211\u4eec\u770b\u770b\u6bcf\u4e2a Hessian \u77e9\u9635\u662f\u4ec0\u4e48\u7c7b\u578b\u7684\u77e9\u9635\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u770b\u770b\u5b83\u662f\u5426\u662f\u6b63\u5b9a\u3001\u8d1f\u5b9a\u3001\u4e0d\u5b9a\u7b49\u3002<\/span>\n<ul>\n<li style=\"margin-left:30px; margin-bottom:10px; margin-top:10px\"><span style=\"color:#262626;font-weight: normal;\">\u5982\u679cHessian\u77e9\u9635\u662f\u6b63\u5b9a\u7684\uff0c\u90a3\u4e48\u4e34\u754c\u70b9\u5c31\u662f\u51fd\u6570\u7684<span style=\"color:#1976d2;\"><strong>\u76f8\u5bf9\u6700\u5c0f\u503c<\/strong><\/span>\u3002<\/span><\/li>\n<li style=\"margin-left:30px; margin-bottom:10px\"><span style=\"color:#262626;font-weight: normal;\">\u5982\u679cHessian\u77e9\u9635\u662f\u8d1f\u5b9a\u7684\uff0c\u90a3\u4e48\u4e34\u754c\u70b9\u5c31\u662f\u51fd\u6570\u7684\u4e00\u4e2a<span style=\"color:#1976d2;\"><strong>\u76f8\u5bf9\u6700\u5927\u503c<\/strong><\/span>\u3002<\/span><\/li>\n<li style=\"margin-left:30px;\"><span style=\"color:#262626;font-weight: normal;\">\u5982\u679cHessian\u77e9\u9635\u672a\u5b9a\u4e49\uff0c\u5219\u4e34\u754c\u70b9\u662f<span style=\"color:#1976d2;\"><strong>\u978d\u70b9<\/strong><\/span>\u3002<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 class=\"wp-block-heading\">\u8ba1\u7b97\u591a\u53d8\u91cf\u51fd\u6570\u7684\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c\u7684\u793a\u4f8b<\/h3>\n<p>\u4e3a\u4e86\u4e86\u89e3\u8fd9\u662f\u5982\u4f55\u5b8c\u6210\u7684\uff0c\u4e0b\u9762\u662f\u4f7f\u7528 Hessian \u77e9\u9635\u8ba1\u7b97\u548c\u5206\u7c7b\u51fd\u6570\u7684\u76f8\u5bf9\u6781\u503c\u7684\u793a\u4f8b\uff1a<\/p>\n<ul>\n<li>\u6c42\u4ee5\u4e0b\u591a\u53d8\u91cf\u51fd\u6570\u7684\u6240\u6709\u76f8\u5bf9\u6781\u503c\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f422cf8ac57ec69b69af091986d534da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)=x^2-y^2+2xy+ 4x-4y\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"262\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u9700\u8981\u505a\u7684\u7b2c\u4e00\u4ef6\u4e8b\u662f\u8ba1\u7b97\u51fd\u6570\u7684\u96c5\u53ef\u6bd4\u77e9\u9635\uff0c\u5728\u672c\u4f8b\u4e2d\u5b83\u5c06\u4e0e\u68af\u5ea6\u4e00\u81f4\uff0c\u56e0\u4e3a\u5b83\u662f\u4e00\u4e2a\u6807\u91cf\u51fd\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b583925d8f0f0a93afa22c409ae00aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\nabla f (x,y)=(2x+2y+4 \\ , \\ -2y+2x-4 )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u5fc5\u987b\u786e\u5b9a\u4e34\u754c\u70b9\uff0c\u4e3a\u6b64\uff0c\u6211\u4eec\u5c06\u83b7\u5f97\u7684\u65b9\u7a0b\u7b49\u4e8e 0\uff0c\u5e76\u6c42\u89e3\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b72f879adb6df18c19610c21eba3887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\nabla f (x,y)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d851eb626a9bd385aec8f68c9df71a39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left. \\begin{array}{l} 2x+2y+4 =0 \\\\[2ex] -2y+2x-4=0 \\end{array}\\right\\} \\longrightarrow \\left. \\begin{array}{c} x = 0 \\\\[1.1ex] y = -2 \\end{array}\\right\\} \\longrightarrow \\ (0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"383\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6240\u4ee5\u6211\u4eec\u627e\u5230\u7684\u4e34\u754c\u70b9\u662f(0,-2)\u3002<\/p>\n<p>\u4e00\u65e6\u627e\u5230\u51fd\u6570\u7684\u4e34\u754c\u70b9\uff0c\u6211\u4eec\u5c31\u5fc5\u987b\u8ba1\u7b97Hessian\u77e9\u9635\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-702fa5f5c3e3d872e1ec0dad0e3216c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (x,y)=\\begin{pmatrix}2 &amp; 2 \\\\[1.1ex] 2 &amp; -2  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"160\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u5e76\u4e14\uff0c\u663e\u7136\uff0c\u5728\u4e34\u754c\u70b9\u8bc4\u4f30\u7684 Hessian \u77e9\u9635\u662f\u76f8\u540c\u7684\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b56ffff28d1a9b98c9848891ae924eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (0,-2)=\\begin{pmatrix}2 &amp; 2 \\\\[1.1ex] 2 &amp; -2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u4e3a\u4e86\u4e86\u89e3\u5b83\u662f\u4ec0\u4e48\u7c7b\u578b\u7684\u77e9\u9635\uff0c\u6211\u4eec\u5c06\u4f7f\u7528\u897f\u5c14\u7ef4\u65af\u7279\u51c6\u5219\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u6c42\u89e3\u77e9\u9635\u7684\u4e3b\u8981\u6b21\u8981\u9879\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-226cf6a18d27a9da300823c13158d56a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}2 \\end{vmatrix} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"50\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d66efe9fca481475009bb1703939e4f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}2 &amp; 2 \\\\[1.1ex] 2 &amp; -2 \\end{vmatrix} = -8\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"104\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u4e3b\u6b21\u6570 1\uff08\u5947\u6570\uff09\u4e3a\u6b63\uff0c\u4e3b\u6b21\u6570 2\uff08\u5076\u6570\uff09\u4e3a\u8d1f\uff0c\u56e0\u6b64\u6839\u636e\u897f\u5c14\u7ef4\u65af\u7279\u51c6\u5219\uff0c\u5b83\u662f\u4e00\u4e2a<strong>\u4e0d\u5b9a\u77e9\u9635\u3002<\/strong>\u56e0\u6b64\uff0c\u4e34\u754c\u70b9 (0,-2) \u662f<strong>\u978d\u70b9\u3002<\/strong> <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\">\u4f7f\u7528 Hessian \u77e9\u9635\u786e\u5b9a\u51fd\u6570\u7684\u51f9\u6027\u6216\u51f8\u6027<\/h2>\n<p>Hessian\u77e9\u9635\u7684\u53e6\u4e00\u4e2a\u7528\u9014\u662f\u77e5\u9053\u4e00\u4e2a\u51fd\u6570\u662f\u51f9\u51fd\u6570\u8fd8\u662f\u51f8\u51fd\u6570\u3002\u8fd9\u53ef\u4ee5\u6839\u636e\u4ee5\u4e0b\u5b9a\u7406\u6765\u786e\u5b9a\uff1a<\/p>\n<p>\u662f<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e87f01194fba5ba72beb64431139ece0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A \\subseteq \\mathbb{R}^n\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -3px;\"><\/p>\n<p>\u4e00\u4e2a\u5f00\u96c6\u548c<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4fa98607bbf0ec2af91778a78a134c97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f \\colon A \\to \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u4e8c\u9636\u5bfc\u6570\u8fde\u7eed\u7684\u51fd\u6570\uff0c\u5176\u51f9\u51f8\u7531 Hessian \u77e9\u9635\u5b9a\u4e49\uff1a<\/p>\n<ul style=\"color:#1976d2; font-weight: bold;\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">\u529f\u80fd\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span>\u6574\u4f53\u5448<span style=\"color:#1976d2;\"><strong>\u51f8\u5f62<\/strong><\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u5f53\u4e14\u4ec5\u5f53\u5176 Hessian \u77e9\u9635\u5728\u96c6\u5408\u7684\u6bcf\u4e2a\u70b9\u90fd\u662f\u534a\u6b63\u5b9a\u7684\u3002<\/li>\n<li><span style=\"color:#262626;font-weight: normal;\">\u529f\u80fd\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span>\u6574\u4e2a\u8fc7\u7a0b\u90fd\u662f<span style=\"color:#1976d2;\"><strong>\u4e25\u683c\u51f8\u7684<\/strong><\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u5f53\u4e14\u4ec5\u5f53\u5176 Hessian \u77e9\u9635\u5728\u96c6\u5408\u7684\u6bcf\u4e2a\u70b9\u90fd\u662f\u6b63\u5b9a\u7684\u3002<\/li>\n<\/ul>\n<p> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong>\u56e0\u6b64\uff0c\u5982\u679c<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u662f\u96c5\u53ef\u6bd4\u77e9\u9635\u4e5f\u6d88\u5931\u7684\u70b9\u5904\u7684\u51f8\u51fd\u6570\uff0c\u8be5\u70b9\u662f<strong>\u5c40\u90e8\u6700\u5c0f\u503c<\/strong>\u3002<\/p>\n<ul style=\"color:#1976d2; font-weight: bold;\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">\u529f\u80fd\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span>\u6574\u4f53\u5448<span style=\"color:#1976d2;\"><strong>\u51f9\u5f62<\/strong><\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u5f53\u4e14\u4ec5\u5f53\u5176 Hessian \u77e9\u9635\u5728\u96c6\u5408\u7684\u6bcf\u4e2a\u70b9\u90fd\u662f\u8d1f\u534a\u5b9a\u7684\u3002<\/li>\n<li><span style=\"color:#262626;font-weight: normal;\">\u529f\u80fd\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span>\u6574\u4f53\u662f<span style=\"color:#1976d2;\"><strong>\u4e25\u683c\u51f9\u7684<\/strong><\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u5f53\u4e14\u4ec5\u5f53\u5176 Hessian \u77e9\u9635\u5728\u96c6\u5408\u7684\u6bcf\u4e2a\u70b9\u90fd\u662f\u8d1f\u5b9a\u7684\u3002<\/li>\n<\/ul>\n<p> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong>\u56e0\u6b64\uff0c\u5982\u679c<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u662f\u96c5\u53ef\u6bd4\u77e9\u9635\u4e5f\u6d88\u5931\u7684\u70b9\u5904\u7684\u51f9\u51fd\u6570\uff0c\u8be5\u70b9\u662f<strong>\u5c40\u90e8\u6700\u5927\u503c<\/strong>\u3002<\/p>\n<p>\u4e0b\u9762\u662f\u4e00\u4e2a\u5728 3 \u7ef4\u7a7a\u95f4\u4e2d\u8868\u793a\u7684\u51f8\u51fd\u6570\u548c\u53e6\u4e00\u4e2a\u51f9\u51fd\u6570\u7684\u793a\u4f8b\uff1a <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-7\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\">\u51f8\u51fd\u6570<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/convexite-et-concavite-d-une-fonction-avec-la-matrice-hessienne.webp\" alt=\"\u7528 Hessian \u77e9\u9635\u786e\u5b9a\u51f8\u51fd\u6570\u6216\u51f9\u51fd\u6570\" width=\"411\" height=\"308\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"><strong>\u51f9\u51fd\u6570<\/strong><\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/concavite-et-convexite-d-une-fonction-avec-la-matrice-hessienne.webp\" alt=\"\u56fe\u50cf\u7684 ALT \u5c5e\u6027\u4e3a\u7a7a\uff1b\u5b83\u7684\u6587\u4ef6\u540d\u662fconcavity-and-convexity-of-a-function-with-the-hessian-matrix-1024x768.jpg\" width=\"411\" height=\"307\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Hessian\u77e9\u9635\u7684\u66f4\u591a\u5e94\u7528<\/h2>\n<p>Hessian \u77e9\u9635\u7684\u4e3b\u8981\u7528\u9014\u662f\u6211\u4eec\u5df2\u7ecf\u770b\u5230\u7684\uff0c\u4f46\u662f\uff0c\u5b83\u8fd8\u6709\u5176\u4ed6\u5e94\u7528\u3002\u6211\u4eec\u5728\u4e0b\u9762\u4e3a\u6700\u597d\u5947\u7684\u4eba\u89e3\u91ca\u5b83\u4eec\u3002<\/p>\n<h3 class=\"wp-block-heading\">\u6cf0\u52d2\u591a\u9879\u5f0f<\/h3>\n<p>2 \u4e2a\u6216\u66f4\u591a\u53d8\u91cf\u51fd\u6570\u5728\u8be5\u70b9\u7684<strong>\u6cf0\u52d2\u591a\u9879\u5f0f<\/strong>\u5c55\u5f00<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u50cf\u8fd9\u6837\u5f00\u59cb\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-813a551888f2fe0d61201e11c9cf83da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  T(x) = f(a) + (x-a)^T \\nabla f(a) + \\frac{1}{2}(x-a)^T \\operatorname{H}_f(a)(x-a) + \\ldots\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"475\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u6b63\u5982\u60a8\u6240\u770b\u5230\u7684\uff0c\u6cf0\u52d2\u5c55\u5f00\u5f0f\u7684\u4e8c\u9636\u9879\u7531\u5728\u591a\u9879\u5f0f\u5c55\u5f00\u70b9\u6c42\u503c\u7684 Hessian \u77e9\u9635\u7ed9\u51fa\u3002<\/p>\n<h3 class=\"wp-block-heading\">\u5e26\u8fb9\u6846\u7684\u7c97\u9ebb\u5e03\u77e9\u9635<\/h3>\n<p>Hessian\u77e9\u9635\u7684\u53e6\u4e00\u4e2a\u7528\u9014\u662f\u8ba1\u7b97\u591a\u5143\u51fd\u6570\u7684\u6700\u5c0f\u503c\u548c\u6700\u5927\u503c<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9f627258270ef54673906bdea5bc47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x,y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u88ab\u9650\u5236\u4e3a\u53e6\u4e00\u4e2a\u89d2\u8272<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-629bf228b3d2b0003d598c1591ec6000_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x,y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"49\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f7f\u7528<strong>\u6709\u754cHessian\u77e9\u9635<\/strong>\uff0c\u5e76\u9075\u5faa\u4ee5\u4e0b\u8fc7\u7a0b\uff1a<\/p>\n<p><strong>\u6b65\u9aa41\uff1a<\/strong>\u8ba1\u7b97\u62c9\u683c\u6717\u65e5\u51fd\u6570\uff0c\u5176\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7253e3e437e00468c7dc9b5e4546991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle L(x,y,\\lambda) = f(x,y)+ \\lambda \\cdot g(x,y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p><strong>\u6b65\u9aa42\uff1a<\/strong>\u627e\u5230\u62c9\u683c\u6717\u65e5\u51fd\u6570\u7684\u4e34\u754c\u70b9\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u8ba1\u7b97\u62c9\u683c\u6717\u65e5\u51fd\u6570\u7684\u68af\u5ea6\uff0c\u5c06\u65b9\u7a0b\u8bbe\u7f6e\u4e3a 0 \u5e76\u6c42\u89e3\u65b9\u7a0b\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-372152b499747854f2da5a2c8c211ce4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\nabla L = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d76d99126f67279a1302a805e2b12e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial L}{\\partial x} = 0 \\qquad \\cfrac{\\partial L}{\\partial y}=0 \\qquad \\cfrac{\\partial L}{\\partial \\lambda}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"240\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p><strong>\u6b65\u9aa4 3\uff1a<\/strong>\u5bf9\u4e8e\u6211\u4eec\u627e\u5230\u7684\u6bcf\u4e2a\u70b9\uff0c\u6211\u4eec\u8ba1\u7b97\u6709\u754c Hessian\uff0c\u5b83\u7531\u4ee5\u4e0b\u77e9\u9635\u5b9a\u4e49\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d1b2b04de9559a521e6704151c27bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H(f,g) = \\begin{pmatrix}0 &amp; \\cfrac{\\partial g}{\\partial x_1} &amp; \\cfrac{\\partial g}{\\partial x_2} &amp; \\cdots &amp; \\cfrac{\\partial g}{\\partial x_n} \\\\[4ex] \\cfrac{\\partial g}{\\partial x_1} &amp; \\cfrac{\\partial^2 f}{\\partial x_1^2} &amp; \\cfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_2} &amp; \\cdots &amp; \\cfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_n} \\\\[4ex] \\cfrac{\\partial g}{\\partial x_2} &amp; \\cfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_1} &amp; \\cfrac{\\partial^2 f}{\\partial x_2^2} &amp; \\cdots &amp; \\cfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_n} \\\\[3ex] \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\[3ex] \\cfrac{\\partial g}{\\partial x_n} &amp; \\cfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_1} &amp; \\cfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_2} &amp; \\cdots &amp; \\cfrac{\\partial^2 f}{\\partial x_n^2}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"289\" width=\"415\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p><strong>\u6b65\u9aa4 4\uff1a<\/strong>\u6211\u4eec\u786e\u5b9a\u6bcf\u4e2a\u4e34\u754c\u70b9\u662f\u6700\u5927\u503c\u8fd8\u662f\u6700\u5c0f\u503c\uff1a<\/p>\n<ul>\n<li>\u8fd9\u5c06\u662f\u51fd\u6570\u7684\u5c40\u90e8\u6700\u5927\u503c\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u5728\u529f\u80fd\u9650\u5236\u4e0b<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u5982\u679c\u5728\u4e34\u754c\u70b9\u8bc4\u4f30\u7684\u6709\u754c Hessian \u77e9\u9635\u7684\u6700\u540e<em>nm<\/em> \uff08\u5176\u4e2d<em>n<\/em>\u662f\u53d8\u91cf\u6570\uff0c <em>m<\/em>\u662f\u7ea6\u675f\u6570\uff09\u4e3b\u6b21\u6570\u5177\u6709\u4ece\u8d1f\u53f7\u5f00\u59cb\u7684\u4ea4\u66ff\u7b26\u53f7\u3002<\/li>\n<\/ul>\n<ul>\n<li>\u8fd9\u5c06\u662f\u51fd\u6570\u7684\u5c40\u90e8\u6700\u5c0f\u503c\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u5728\u529f\u80fd\u9650\u5236\u4e0b<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u5982\u679c\u5728\u4e34\u754c\u70b9\u8bc4\u4f30\u7684\u5c16\u9510 Hessian \u77e9\u9635\u7684\u6700\u540e<em>nm<\/em> \uff08\u5176\u4e2d<em>n<\/em>\u662f\u53d8\u91cf\u6570\uff0c <em>m<\/em>\u662f\u7ea6\u675f\u6570\uff09\u4e3b\u6b21\u6570\u90fd\u5177\u6709\u8d1f\u53f7\u3002<\/li>\n<\/ul>\n<p>\u5e94\u8be5\u8bb0\u4f4f\uff0c\u4e00\u4e2a\u53d7\u7ea6\u675f\u51fd\u6570\u76f8\u5bf9\u4e8e\u53e6\u4e00\u4e2a\u53d7\u7ea6\u675f\u51fd\u6570\u7684\u76f8\u5bf9\u6700\u5c0f\u503c\u6216\u6700\u5927\u503c\u5bf9\u4e8e\u4e0d\u53d7\u7ea6\u675f\u51fd\u6570\u6765\u8bf4\u4e0d\u4e00\u5b9a\u5982\u6b64\u3002\u56e0\u6b64\uff0c\u6709\u8fb9\u754c\u7684 Hessian \u77e9\u9635\u4ec5\u5bf9\u6b64\u7c7b\u95ee\u9898\u6709\u7528\u3002<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\">\u4e0e\u5176\u4ed6\u4e1a\u52a1\u7684\u5173\u7cfb<\/h3>\n<p>\u6700\u540e\uff0cHessian\u77e9\u9635\u8fd8\u4e0e\u5176\u4ed6\u91cd\u8981\u7684\u8fd0\u7b97\u6216\u77e9\u9635\u8054\u7cfb\u5728\u4e00\u8d77\uff0c\u4e3b\u8981\u662f\u4e0e\u96c5\u53ef\u6bd4\u77e9\u9635\u548c\u4e0e\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u3002<\/p>\n<h4 class=\"wp-block-heading\">\u4e0e\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u5173\u7cfb<\/h4>\n<p>\u51fd\u6570\u7684 Hessian \u77e9\u9635<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u662f\u540c\u4e00\u51fd\u6570\u68af\u5ea6\u7684<strong>\u96c5\u53ef\u6bd4\u77e9\u9635<\/strong>\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96ab1054f3c447eedac17f9ce04b4606_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f = J(\\nabla f)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"97\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<h4 class=\"wp-block-heading\">\u62c9\u666e\u62c9\u65af\u7b97\u5b50<\/h4>\n<p>Hessian \u77e9\u9635\u8ff9\u7b49\u4ef7\u4e8e<strong>\u62c9\u666e\u62c9\u65af\u7b97\u5b50<\/strong>\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77f16c5285ca6a9ac5899d6e832e9a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle tr( H_f) = \\Delta f\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"101\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p>\u8fd9\u4e2a\u7b49\u5f0f\u5f88\u5bb9\u6613\u8bc1\u660e\uff0c\u56e0\u4e3a\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u7684\u5b9a\u4e49\u662f\u51fd\u6570\u68af\u5ea6\u7684\u6563\u5ea6\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5a3e06f41c5bd51c969601d46507a9c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\Delta f =\\nabla \\cdot (\\nabla f) = (\\nabla \\cdot \\nabla )f = \\nabla^2 f\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"262\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\u5176\u8868\u8fbe\u5f0f\u4e3a\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7e1ec07ef9178ea4dd56d9cae72d275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\Delta f = \\cfrac{\\partial ^2 f}{\\partial^2 x_1} +\\cfrac{\\partial ^2 f}{\\partial^2 x_2} + \\cfrac{\\partial ^2 f}{\\partial^2 x_3}+ \\ldots +\\cfrac{\\partial ^2 f}{\\partial^2 x_n}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"310\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p>\u800c\u8fd9\u4e2a\u548c\u53ea\u662fHessian\u77e9\u9635\u7684\u8ff9\uff0c\u56e0\u6b64\u7b49\u4ef7\u6027\u5f97\u5230\u8bc1\u660e\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u9875\u65e0\u7591\u662f\u73b0\u6709\u7684 Hessian \u77e9\u9635\u6700\u5b8c\u6574\u7684\u89e3\u91ca\u3002\u8fd9\u91cc\u89e3\u91ca\u4e86Hessian\u77e9\u9635\u7684\u6982\u5ff5\uff0c\u5982\u4f55\u7528\u4f8b\u5b50\u8ba1\u7b97\u5b83\uff0c\u751a\u81f3 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e7%b2%97%e9%ba%bb%e5%b8%83%e7%9f%a9%e9%98%b5-%e7%b2%97%e9%ba%bb%e5%b8%83-2x2-3x3\/\"> <span class=\"screen-reader-text\">Hessian \u77e9\u9635\uff08\u6216 hesse\uff09<\/span> \u67e5\u770b\u5168\u6587 &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[7],"tags":[],"class_list":["post-313","post","type-post","status-publish","format-standard","hentry","category-7"],"yoast_head":"<!-- This site is optimized 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