# Counterexample to Sylvester's criterion for non-hermitian matrix

Sylvester's criterion is stated and proved here. I was wondering: why is it stated only for Hermitian matrix? If $A$ is not hermitian, does it hold for A? If not, which implication(s) fail and can you provide a counterexample? And if not, is there any adjustment to be made to the proof to make it work?

Consider $$\begin{pmatrix} 1 & 1 \\ -100 & 1\end{pmatrix}$$
• Indeed, for $(1,0)*A*(1,0)^T=1-100=-99$, yet the minors are positive. Jul 8 '15 at 18:40