Hessian matrix

The matrix of all second partial derivatives -- the quadratic term in a function's Taylor expansion.

The Hessian matrix $H_f$ of a scalar function $f$ is the symmetric matrix of second partials, $(H_f)_{ij}=\partial^2f/\partial x_i\partial x_j$ (equal by Clairaut's theorem when $f$ is $C^2$ ). It's the bilinear form that appears as the quadratic term in $f$ 's second-order Taylor approximation, and its eigenvalues classify a critical point as a min, max, or saddle.

Examples

For $f(x,y)=x^3-3xy^2$ , $H_f = \begin{pmatrix}6x & -6y\\ -6y & -6x\end{pmatrix}$ .