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According to wikipedia, a totally positive matrix (https://en.wikipedia.org/wiki/Totally_positive_matrix) always have non-negative eigenvalues. But, is it necessary for a matrix to be totally positive to have non-negative eigenvalues?

I'm looking for the necessary and sufficient condition for a real square matrix (not necessarily symmetric) to have non-negative eigenvalues.

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  • $\begingroup$ You may want to look up diagonal dominance and Gershgorin's circle theorems. $\endgroup$ – bcf Jun 3 '16 at 2:18
  • $\begingroup$ @bcf : Thank you. Gershgorin's circle theorem looks like a good starting point. $\endgroup$ – Sachin Jun 3 '16 at 3:00
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No it is not necessary. Just consider the matrix $$ A= \begin{bmatrix} 2 & -1\\ -1 & 2 \end{bmatrix} $$ which is not totally positive but it is still has non-negative eigenvalues $\{1,3\}.$

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  • $\begingroup$ Is there any necessary condition then? $\endgroup$ – Sachin Jun 3 '16 at 5:35
  • $\begingroup$ Yes, you can get many necessary conditions. e.g. for a square matrix $A,$ $det(A)\geq 0$ or $Tr(A) \geq 0$ are necessary conditions (although these are not sufficient) $\endgroup$ – A. Ray Jun 3 '16 at 5:44

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