Hessian

The $n\times n$ Hessian Matrix of $f$ is given by

$$\left[\begin{array}{ccc}\frac{{\partial }^{2}f}{\partial x_1\partial x_1}& \cdots & \frac{{\partial }^{2}f}{\partial x_1\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{{\partial }^{2}f}{\partial x_n\partial x_1}& \cdots & \frac{{\partial }^{2}f}{\partial x_n\partial x_n}\end{array}\right]$$

where

• $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a real valued function
• The Hessian can be written as $J(\nabla f(x){)}^{†}$, since it is the Jacobian of the Gradient.

Properties

• The Hessian is a symmetric Matrix