# Are all symmetric non-singular matrices positive semi-definite?

Are all symmetric non-singular matrices positive semi-definite? Or are they positive definite, or could they be any thing?

• No, take $-I$. This is symmetric, non singular, but far from positive definite. Take $\operatorname{diag} (1,-1)$ for a better example, which is not definite at all. – copper.hat Oct 3 '15 at 2:35
• A definite example of a non-definite matrix! – Gerry Myerson Oct 3 '15 at 2:58
• Symmetric matrices correspond to quadratic forms, which are certainly not all positive (semi-)definite. – amd Oct 3 '15 at 3:02