Hessian matrix

Matrix of second partial derivatives of a scalar function

Hessian matrix

A Hessian matrix of a function $f:U\to \mathbb{R}$ (with $U\subseteq \mathbb{R}^n$ ) at a point $a\in U$ is the $n\times n$ matrix

Hf(a)=\left(\frac{\partial^2 f}{\partial x_i\,\partial x_j}(a)\right)_{1\le i,j\le n},

provided these second-order partial derivatives exist.

The off-diagonal entries are mixed partial derivatives . Under the hypotheses of the Schwarz–Clairaut theorem , the Hessian is symmetric. The Hessian is used in second derivative tests for classifying critical points .

Examples:

For $f(x,y)=x^2+xy+y^2$ , one has
$Hf(x,y)=\begin{pmatrix}2&1\\[2pt]1&2\end{pmatrix}.$
For $f(x,y,z)=x^2+y^2+z^2$ , one has $Hf(x,y,z)=2I_3$ .