对于一个$n$元函数$f(x_1, x_2, \dots, x_n)$，如果其二阶偏导数都存在，海森矩阵$H$定义为：
$$H = \begin{bmatrix}\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\\frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\\vdots & \vdots & \ddots & \vdots \\\frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2}\end{bmatrix}$$
即： $H_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}$
也就是说，这个矩阵记录了多元函数的二阶偏导数，描述了函数的梯度变化率或者说曲率。