「高等數(shù)學(xué)」雅可比矩陣和黑塞矩陣的異同
雅可比矩陣，Jacobi matrix 或者 Jacobian，是向量值函數(shù)（ f : R n → R m f:\mathbb{R}^n \to \mathbb{R}^m f:Rn→Rm）的一階偏導(dǎo)數(shù)按行排列所得的矩陣。
黑塞矩陣，又叫海森矩陣，Hesse matrix，是多元函數(shù)（ f : R n → R f:\mathbb{R}^n \to \mathbb{R} f:Rn→R）的二階偏導(dǎo)數(shù)組成的方陣。
1、雅可比矩陣 J m × n J_{m\times n} Jm×n?
雅可比矩陣通常是一個mxn的矩陣。
給出一個向量值函數(shù)： h ( x ) = ( h 1 ( x ) , h 2 ( x ) , ? ? , h m ( x ) ) T h(\mathbf{x}) = (h_1(\mathbf{x}),h_2(\mathbf{x}),\cdots,h_m(\mathbf{x}))^T h(x)=(h1?(x),h2?(x),?,hm?(x))T
它的雅可比矩陣是：
J = [ ? h ? x 1 ? ? h ? x n ] = [ ? h 1 ? x 1 ? ? h 1 ? x n ? ? ? ? h m ? x 1 ? ? h m ? x n ] {\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {h} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {h} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial h_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial h_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial h_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial h_{m}}{\partial x_{n}}}\end{bmatrix}}} J=[?x1??h?????xn??h??]= ??x1??h1????x1??hm????????xn??h1????xn??hm??? ?
矩陣的每一行相當(dāng)于每個向量值函數(shù)的分量的梯度的轉(zhuǎn)置，或者叫一階偏導(dǎo)數(shù)按行（row）排列。
一個n元實值函數(shù)的梯度的雅可比矩陣：
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{\displaystyle \mathbf {J} = D[\nabla f(\mathbf{x})] = {\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}\\ \\{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}\\ \\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}&{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}\,}
J=D[?f(x)]=
??x12??2f??x1??x2??2f???x1??xn??2f???x2??x1??2f??x22??2f???x2??xn??2f????????xn??x1??2f??xn??x2??2f???xn2??2f??
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2、黑塞矩陣 H n × n H_{n\times n} Hn×n?
黑塞矩陣一定是一個方陣。
二階混合偏導(dǎo)數(shù)：
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\frac{\partial^2 f}{\partial y \, \partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = f_{xy}
?y?x?2f?=?y??(?x?f?)=fxy?
對于一個n元實值函數(shù)
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f(x)，它的梯度為一個列向量：
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\nabla f(\mathbf{x}) = (f_{x_1}(\mathbf{x}),f_{x_2}(\mathbf{x}),\cdots,f_{x_n}(\mathbf{x}))^T
?f(x)=(fx1??(x),fx2??(x),?,fxn??(x))T
對其求二階偏導(dǎo)數(shù)，并將偏導(dǎo)數(shù)按列（col）排列。
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{\displaystyle \mathbf {H} ={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}\,}
H=
??x12??2f??x2??x1??2f???xn??x1??2f???x1??x2??2f??x22??2f???xn??x2??2f????????x1??xn??2f??x2??xn??2f???xn2??2f??
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因此：
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對于一個二階可微的n元實值函數(shù)，它的黑塞矩陣的轉(zhuǎn)置??它的梯度的雅可比矩陣。
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對于一個二階連續(xù)可微的n元實值函數(shù)，其二階混合偏導(dǎo)數(shù)： ? 2 f ? y ? ? x = ? 2 f ? x ? ? y \frac{\partial^2 f}{\partial y \, \partial x} = \frac{\partial^2 f}{\partial x \, \partial y} ?y?x?2f?=?x?y?2f?。此時，其黑塞矩陣??它的梯度的雅可比矩陣。文章來源：http://www.zghlxwxcb.cn/news/detail-695841.html
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在很多地方，遇到的都是二階連續(xù)可微的情況，因此有些地方對雅可比矩陣和黑塞矩陣不加以區(qū)分。文章來源地址http://www.zghlxwxcb.cn/news/detail-695841.html
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