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；