Jacobian matrix （雅可比矩阵或者是Jacobian）

Jacobian是向量函数f(x)f(\mathbf{x})相对于向量x\mathbf{x}的偏导数以一定方式排列成的矩阵

1.向量函数:f(x)f(\mathbf{x})

f(x)=⎡⎣⎢⎢⎢⎢⎢⎢f1(x)...fi(x)...fn(x)⎤⎦⎥⎥⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢⎢⎢f1(x1,...,xm)...fi(x1,...,xm)...fn(x1,...,xm)⎤⎦⎥⎥⎥⎥⎥⎥n∗1f(\mathbf{x})=\begin{bmatrix}f_{1}(\mathbf{x})\\ ...
\\ f_{\mathit{i}}(\mathbf{x})\\ ...\\ f_{\mathit{n}}(\mathbf{x})
\end{bmatrix}=\begin{bmatrix}f_{1}(x_{1},...,x_{m})\\ ...\\ f_{\mathit{i}}(x_{1},...,x_{m})\\ ...\\ f_{\mathit{n}}(x_{1},...,x_{m})\end{bmatrix}_{n*1}

其中：

fi(x1,...,xm)f_{\mathit{i}}(x_{1},...,x_{m})是向量x\mathbf{x}的函数；x\mathbf{x}是一个向量，含有mm个参数：x=(x1,...,xm)\mathbf{x}=(x_{1},...,x_{m})；

2.Jacobian:J(x1,...,xm)\boldsymbol{\mathbf{\mathit{J}}}(x_{1},...,x_{m})

J(x1,...,xm)=∂f∂x=∂(f1,..,fn)∂(x1,..,xm)T=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢∂f1∂x1∂fi∂x1∂fn∂x1.........∂f1∂xm∂fi∂xm∂fn∂xm⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥n∗m\boldsymbol{\mathbf{\mathit{J}}}(x_{1},...,x_{m})=\frac{\partial \mathbf{f}}{\partial \mathbf{x}}=\frac{\partial (f_{1},..,f_{n})}{\partial (x_{1},..,x_{m})^{T}}=\begin{bmatrix} &\frac{\partial f_{1}}{\partial x_{1}} &... &\frac{\partial f_{1}}{\partial x_{m}} \\ & & & \\ &\frac{\partial f_{i}}{\partial x_{1}} &... &\frac{\partial f_{i}}{\partial x_{m}} \\ & & & \\ &\frac{\partial f_{n}}{\partial x_{1}} &... &\frac{\partial f_{n}}{\partial x_{m}} \end{bmatrix}_{n*m}