Counterexample to Sylvester's criterion for non-hermitian matrix

Question

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?

score 1 · Accepted Answer · answered Jul 08 '15 at 18:15

1

Consider $$\begin{pmatrix} 1 & 1 \\ -100 & 1\end{pmatrix}$$

answered Jul 08 '15 at 18:15

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Indeed, for $(1,0)A(1,0)^T=1-100=-99$, yet the minors are positive. – MickG Jul 08 '15 at 18:40

Counterexample to Sylvester's criterion for non-hermitian matrix

1 Answers1