3.7. Hessians

[!note] Hessian

The Hessian of a scalar-valued function $f:{\mathbb{R}}^n\to \mathbb{R}$ is the square matrix of second-order partial derivatives of $f$ with respect to the components of its input vector $x=[x_1,x_2,\dots ,x_n{]}^T$. It characterizes the local curvature of $f$ at a given point $x$ and is defined as
$H_f={\nabla }^{2}f(x)=\left[\begin{array}{ccc}\frac{{\partial }^{2}f}{\partial x_1^2}& \cdots & \frac{{\partial }^{2}f}{\partial x_1\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{{\partial }^{2}f}{\partial x_n\partial x_1}& \cdots & \frac{{\partial }^{2}f}{\partial x_n^2}\end{array}\right]\in {\mathbb{R}}^{n\times n}.$
The Hessian indicates the directions and magnitudes of curvature, determining whether $f$ curves upward, downward, or remains flat near $x$.

Properties

(Magnitude)

(Eigendecomposition)