Is there a Sylvester's criterion for negative semidefinite matrices? I suspect such a criterion to be:

  • All principal minors with odd dimension are non-positive.
  • All principal minors with even dimension are non-negative.

Please include a reference if it exists.

  • 2
    $\begingroup$ This is indeed the case. In order to prove it, simply apply the usual criterion to $-A$, where $A$ is positive semidefinite. I have not seen this in a linear algebra text, but I know that this comes up in multivariate optimization problems (in the "second derivative test"). $\endgroup$ – Omnomnomnom Dec 10 '15 at 12:48
  • 3
    $\begingroup$ This is treated in Simon & Blume's Mathematics for Economists, page 383. $\endgroup$ – Nigel Overmars Dec 10 '15 at 12:54
  • $\begingroup$ For a quick numerical check, it is also equivalent to: all the eigenvalues are $\le 0$. Or, the coefficients of the characteristic polynomial have the same sign ( think $a(x+1)(x+2) \ldots $ ). $\endgroup$ – Orest Bucicovschi Dec 10 '15 at 13:32

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