Positive definiteness

Positive definiteness

Positive definiteness is a structural positivity condition on matrices, kernels or functions.

For a real symmetric matrix $M$, positive semidefiniteness means $\mathbf c^T M\mathbf c\ge 0$ for every vector $\mathbf c$. For translation-invariant functions, the analogous condition requires every finite matrix of pairwise values to have non-negative quadratic form.

Bochner's theorem connects continuous positive-definite functions with Fourier transforms of finite non-negative measures.

For a fuller introduction, see What is positive definiteness?. .