Hessian Matrix

Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is a twice partially differentiable function. The Hessian matrix of $f$ is the matrix of second partial derivatives.

$$H(f) = D^2 f = \begin{pmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_2 \partial x_1} & \cdots & \frac{\partial^2 f}{\partial x_n \partial x_1} \\ \frac{\partial^2 f}{\partial x_1 \partial x_2} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_n \partial x_2} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_1 \partial x_n} & \frac{\partial^2 f}{\partial x_2 \partial x_n} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{pmatrix}$$

At a point $x \in \mathbb{R}^n$ is the $n \times n$ matrix of second partial derivatives of $f$ at $x$. Denoted by $H(f)(x)$.