1
$\begingroup$

I know that for a given symmetric matrix $\mathbf{A}$ the matrix is positive definite / negative definite $\Longleftrightarrow$ main minors of $\mathbf{A}$ / $-\mathbf{A}$ are positive.

Is it also true that for the case where all main minors are positive or zero, that I can conclude the semi definiteness? I could only find statements for the case in which all minors are positive.

$\endgroup$
4
$\begingroup$

No, one cannot show semidefiniteness in that way. Consider the case where all main minors are zero. Does that mean that $A$ is positive semidefinite? Or is it $-A$?

$$A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$

is indefinite and has all main minors zero.

$\endgroup$
1
  • $\begingroup$ Thank you for your quick answer. I allready thought that this would not work, but I couldn’t come up with a counterexample. $\endgroup$ – MrYouMath Sep 8 '17 at 17:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.