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Suppose we have a matrix $C = DP$ where D is a real diagonal matrix with all the entries positive, and P is a real matrix whose eigen values are positive and all the elements are positive. Moreover, each row of P sums to the same value. Can we conclude that C has all the eigen values positive?

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  • $\begingroup$ If the matrix $P$ is symmetric (in addition to the above conditions), I suspect that $C$ can be shown to have all positive eigenvalues. But I'm not 100% sure. $\endgroup$ – Michael Seifert Nov 28 '18 at 14:01
  • $\begingroup$ @MichaelSeifert Indeed. If $P$ is symmetric, then $C=DP$ is similar to $D^{-1/2}CD^{1/2}=D^{1/2}PD^{1/2}$, which is positive definite. $\endgroup$ – user1551 Nov 29 '18 at 11:41
  • $\begingroup$ Why is $D^{1/2}PD{1/2}$ positive definite if P is symmetric? @user1551 $\endgroup$ – Suhan Shetty Nov 29 '18 at 12:26
  • $\begingroup$ @SuhanShetty In your question, $P$ is assumed to be a real matrix with a positive spectrum. If it is also symmetric, it is positive definite. Hence $D^{1/2}PD^{1/2}$ (which is congruent to $P$) is positive definite too. $\endgroup$ – user1551 Nov 29 '18 at 12:42
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No. Random counterexample: $$ P=\pmatrix{3&2&5\\ 4&5&1\\ 3&3&4},\ D=\operatorname{diag}(8,1,7). $$ The eigenvalues of $P$ are $10,1,1$ but the eigenvalues of $DP$ are $57.05$ and $-0.025\pm3.13i$.

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