Hessian matrix

The matrix of second partial derivatives of a scalar function f:ℝ^k→ℝ.

Hessian matrix

Let $U\subseteq\mathbb{R}^k$ be open and let $f:U\to\mathbb{R}$ . If all second partial derivatives exist at $a\in U$ , the Hessian matrix of $f$ at $a$ is the $k\times k$ matrix

H_f(a) := \left[\frac{\partial^2 f}{\partial x_i\,\partial x_j}(a)\right]_{1\le i,j\le k}.

When $f$ is $C^2$ , the Hessian is symmetric (mixed partials commute) and governs second-order Taylor approximation and second-derivative tests for extrema.

Examples:

If $f(x,y)=x^2+y^2$ , then $H_f(x,y)=\begin{pmatrix}2&0\\0&2\end{pmatrix}$ .
If $f(x,y)=xy$ , then $H_f(x,y)=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ .
For a linear function $f(x)=\langle c,x\rangle$ , the Hessian is the zero matrix.