Hessian matrix ($\mathrm{Hess}f$)

The Hessian is the symmetric matrix of second partial derivatives; definiteness classifies nondegenerate critical points.

For $f(x,y)=g(x)+h(y)$, a degenerate Hessian matrix can be supplemented by the first nonzero term of each one-variable displacement. Two negative leading even powers give a strict local maximum, two positive ones a strict local minimum, and opposite signs give a saddle. Looking along the coordinate directions and bounding the separated terms avoids an inconclusive quadratic test.

A symmetric matrix $A$ is negative semidefinite when $v^{T}Av\le 0$ for every vector $v$. Equivalently, all its eigenvalues are nonpositive.

At an interior local maximum of a twice differentiable function, its Hessian matrix is negative semidefinite; at a local minimum it is positive semidefinite. For a scalar function on Euclidean space, the Laplacian is therefore nonpositive at a local maximum and nonnegative at a local minimum.

A critical point of a twice differentiable function is nondegenerate when its Hessian matrix is invertible there. A positive-definite Hessian gives a strict local minimum, a negative-definite Hessian gives a strict local maximum, and an indefinite Hessian gives a saddle point.