「高等数学」雅可比矩阵和黑塞矩阵的异同
「高等数学」雅可比矩阵和黑塞矩阵的异同
雅可比矩阵，Jacobi matrix 或者 Jacobian，是向量值函数（ f : R n → R m f:\mathbb{R}^n \to \mathbb{R}^m f:Rn→Rm）的一阶偏导数按行排列所得的矩阵。
黑塞矩阵，又叫海森矩阵，Hesse matrix，是多元函数（ f : R n → R f:\mathbb{R}^n \to \mathbb{R} f:Rn→R）的二阶偏导数组成的方阵。
1、雅可比矩阵 J m × n J_{m\times n} Jm×n
雅可比矩阵通常是一个mxn的矩阵。
给出一个向量值函数： 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
它的雅可比矩阵是：
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
矩阵的每一行相当于每个向量值函数的分量的梯度的转置，或者叫一阶偏导数按行（row）排列。
一个n元实值函数的梯度的雅可比矩阵：
J = D [ ∇ f ( x ) ] = [ ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 2 ∂ x 1 ⋯ ∂ 2 f ∂ x n ∂ x 1 ∂ 2 f ∂ x 1 ∂ x 2 ∂ 2 f ∂ x 2 2 ⋯ ∂ 2 f ∂ x n ∂ x 2 ⋮ ⋮ ⋱ ⋮ ∂ 2 f ∂ x 1 ∂ x n ∂ 2 f ∂ x 2 ∂ x n ⋯ ∂ 2 f ∂ x n 2 ] {\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
2、黑塞矩阵 H n × n H_{n\times n} Hn×n
黑塞矩阵一定是一个方阵。
二阶混合偏导数：
∂ 2 f ∂ y ∂ x = ∂ ∂ y ( ∂ f ∂ x ) = f x y \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
对于一个n元实值函数 f ( x ) f(\mathbf{x}) f(x)，它的梯度为一个列向量： ∇ f ( x ) = ( f x 1 ( x ) , f x 2 ( x ) , ⋯ , f x n ( x ) ) T \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
对其求二阶偏导数，并将偏导数按列（col）排列。
H = [ ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ⋯ ∂ 2 f ∂ x 1 ∂ x n ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 ⋯ ∂ 2 f ∂ x 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ 2 f ∂ x n ∂ x 1 ∂ 2 f ∂ x n ∂ x 2 ⋯ ∂ 2 f ∂ x n 2 ] {\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
因此：
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对于一个二阶可微的n元实值函数，它的黑塞矩阵的转置🟰它的梯度的雅可比矩阵。
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对于一个二阶连续可微的n元实值函数，其二阶混合偏导数： ∂ 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。此时，其黑塞矩阵🟰它的梯度的雅可比矩阵。
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在很多地方，遇到的都是二阶连续可微的情况，因此有些地方对雅可比矩阵和黑塞矩阵不加以区分。