positive definite

A positive-definite form is one that behaves like an honest notion of squared length: every nonzero vector has a strictly positive value, and only the zero vector measures zero. This is precisely the property that lets a symmetric form serve as an inner product, giving genuine distances and angles with no degenerate or negative ‘lengths’ lurking.

Over R (or any ordered field), a quadratic form Q is positive definite if Q(v) > 0 for every v ≠ 0; the associated symmetric bilinear form is then an inner product. In signature terms this means (p, q) = (n, 0): all diagonal entries positive, rank full. A symmetric matrix A is positive definite when x^T A x > 0 for all x ≠ 0, equivalently all its eigenvalues are positive.

Several tests detect positive definiteness: Sylvester's criterion asks that every leading principal minor of A be positive; alternatively all eigenvalues positive, or A = P^T P for some invertible P (a Cholesky-type factorization). The weaker notion positive semidefinite allows Q(v) ≥ 0 with equality possible for nonzero v, corresponding to (p, 0) with p < n and a nontrivial radical.