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

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    $\begingroup$ 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. $\endgroup$ – copper.hat Oct 3 '15 at 2:35
  • $\begingroup$ A definite example of a non-definite matrix! $\endgroup$ – Gerry Myerson Oct 3 '15 at 2:58
  • $\begingroup$ Symmetric matrices correspond to quadratic forms, which are certainly not all positive (semi-)definite. $\endgroup$ – amd Oct 3 '15 at 3:02

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