Sylvester's Criterion

Sylvester’s Criterion is about solving a symmetric matrix’s definite-ness.

Given a $n\times n$ symmetric matrix $M$, let $M_k$ be the top-left $k\times k$ sub-matrix. Then the criterion is:

• $M$ is positive definite iff for all $k$, $\det M_k\gt 0$.
• $M$ is negative definite iff for all odd $k$, $\det M_k\lt 0$ and for all even $k$, $\det M_k\gt 0$.
• Otherwise, $M$ is indefinite.