pc网站增加手机站,浙江网站建设多少钱,wordpress入门建站,wordpress文章省略「高等数学」雅可比矩阵和黑塞矩阵的异同
雅可比矩阵#xff0c;Jacobi matrix 或者 Jacobian#xff0c;是向量值函数#xff08; f : R n → R m f:\mathbb{R}^n \to \mathbb{R}^m f:Rn→Rm#xff09;的一阶偏导数按行排列所得的矩阵。
黑塞矩阵#xff0c;又叫海森矩…「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}}\,} JD[∇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 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。 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。 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。
