definiteness

Definiteness asks the simplest possible question about a quadratic form: what signs does Q(v) take as v ranges over nonzero vectors? Positive definite means Q(v) > 0 for every nonzero v (the form is a genuine 'squared length'); negative definite means Q(v) < 0 always; indefinite means Q takes both signs. The bowl, the upside-down bowl, and the saddle are the three pictures to carry.

Two softer cases sit in between. Positive semidefinite means Q(v) >= 0 for all v, allowing zeros along a subspace; negative semidefinite is the mirror. Semidefinite is the borderline where the form bottoms out at zero on a whole direction without dipping below — exactly the situation at a degenerate critical point in optimization.

In terms of the signature (p, q, z), the dictionary is exact: positive definite is (n, 0, 0); negative definite is (0, n, 0); positive semidefinite is (p, 0, z) with q = 0; indefinite is whenever p > 0 and q > 0. Equivalently, definiteness is read straight off the signs of the eigenvalues of the symmetric matrix: all positive, all negative, mixed, or with some zeros.

This is the concept that makes quadratic forms useful in calculus and optimization. The Hessian of a function is a symmetric matrix; its definiteness at a critical point tells you minimum (positive definite), maximum (negative definite), or saddle (indefinite). The semidefinite case is the inconclusive one where the second-derivative test fails and you must look further. Sylvester's criterion gives a determinant test for the positive-definite case.