常用数学矩阵【Jacobi矩阵，Hessan 矩阵】

一、雅可比（Jacobi）矩阵

对于函数

\[y=f(x) \]

其中，\(x=(x_1;x_2,...;x_n)\),\(y=(y_1;y_2;...;y_m)\)
则Jacobi矩阵为：

\[ J= \begin{pmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \frac{\partial y_1}{\partial x_3} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \frac{\partial y_2}{\partial x_3} & \cdots & \frac{\partial y_2}{\partial x_n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \frac{\partial y_m}{\partial x_3} & \cdots & \frac{\partial y_m}{\partial x_n} \\ \end{pmatrix} \]

如果函数在一点\(p\)处可微，则Jacobi矩阵为函数在这一点处的最优线性逼近，即，
\(f(x)\approx f(p)+J(p)(x-p)\)

二、海塞（Hessan）矩阵

对于函数\(f(x)\),其中，\(x=(x_1;x_2;x_3,...;x_n)\),其Hessan 矩阵为：

\[ H= \begin{pmatrix} \frac{\partial f}{\partial x_1\partial x_1} & \frac{\partial f}{\partial x_1\partial x_2} & \cdots & \frac{\partial f}{\partial x_1\partial x_n} \\ \frac{\partial f}{\partial x_2\partial x_1} & \frac{\partial f}{\partial x_2\partial x_2} & \cdots & \frac{\partial f}{\partial x_2\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f}{\partial x_n\partial x_1} & \frac{\partial f}{\partial x_n\partial x_2} & \cdots & \frac{\partial f}{\partial x_n\partial x_n} \\ \end{pmatrix} \]

Hessan matrix和Jacobi matrix关系：

\[H_f=J(\nabla f^T) \]

最优化中应用
当Hessan matrix正定时，在这一点取极小值；
当Hessan matrix负定时，在这一点取极大值；

参考

Jacobi matrix
Hessan matrix