Hessian

Suppose $f:{\mathbb{R}}^n\to \mathbb{R}$ is a function taking as input a vector $\mathbf{x}\in {\mathbb{R}}^n$ and outputting a scalar $f(\mathbf{x})\in \mathbb{R}$ If all second-order partial derivatives of $f$ exist, then the Hessian matrix $\mathbf{H}$ of $f$ is a square $n\times n$ matrix, usually defined and arranged as

$${\mathbf{H}}_f=\left[\begin{array}{cccc}\frac{{\partial }^{2}f}{\partial x_1^2}& \frac{{\partial }^{2}f}{\partial x_1\partial x_2}& \cdots & \frac{{\partial }^{2}f}{\partial x_1\partial x_n}\\ \frac{{\partial }^{2}f}{\partial x_2\partial x_1}& \frac{{\partial }^{2}f}{\partial x_2^2}& \cdots & \frac{{\partial }^{2}f}{\partial x_2\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{{\partial }^{2}f}{\partial x_n\partial x_1}& \frac{{\partial }^{2}f}{\partial x_n\partial x_2}& \cdots & \frac{{\partial }^{2}f}{\partial x_n^2}\end{array}\right]$$

Examples

$$H_f(a,b)=\left[\begin{array}{cc}{f}_{xx}& f_{xy}\\ f_{yx}& f_{yy}\end{array}\right]{|}_{(a,b)}$$