Definiteness–Signature Relationship

Definiteness and signature are directly related: the definiteness of a symmetric matrix is completely determined by its signature $(p, z, n)$. The matrix is PD iff $p = n$ (all positive, none zero or negative), PSD iff $n = 0$ (no negative eigenvalues), ND iff $p = 0$ and $z = 0$, NSD iff $p = 0$, and indefinite iff both $p > 0$ and $n > 0$.

This gives you a quick checklist: look at the sign count of eigenvalues. If there is even one negative eigenvalue and one positive eigenvalue, the matrix is indefinite. If there are only non-negatives with at least one zero, it is PSD but not PD. This connection between signature and definiteness makes signature a fundamental classification tool.