Hessian Matrix

Definition: Hessian Matrix

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a twice partially differentiable real scalar field.

The Hessian matrix of $f$ is the $n\times n$-matrix $H_f$ whose columns are the gradients of $f$‘s first order partial derivatives:

$$H_f\stackrel{\text{def}}{=}\left[\begin{array}{ccc}|& |& |\\ \nabla ({\partial }_{1}f)& \cdots & \nabla ({\partial }_{n}f)\\ |& |& |\end{array}\right]=\left[\begin{array}{ccc}{\partial }_1{\partial }_{1}f& \cdots & {\partial }_1{\partial }_{n}f\\ ⋮& \ddots & ⋮\\ {\partial }_n{\partial }_{1}f& \cdots & {\partial }_n{\partial }_{n}f\end{array}\right]$$

NOTATION

The Hessian matrix $H_f$ is different for different $\mathbf{x}\in \mathcal{D}$, since the partial derivatives of $f$ depend on $\mathbf{x}$. The Hessian matrix at a given ${\mathbf{x}}_0$ is thus denoted as $H_f({\mathbf{x}}_0)$ to make this dependency apparent.

Theorem: Symmetry of the Hessian Matrix

Let $f:D\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a twice partially differentiable real scalar field.

If all second partial derivatives of $f$ are also continuous, then the Hessian matrix of $f$ is symmetric for every $\mathbf{x}\in D$.

PROOF

This follows directly from Schwarz’s theorem.