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Let $X$ be a positive definite matrix in $\Bbb R^{n\times n}$ or $\Bbb C^{n\times n}$, is there any known connection between the eigenvalues of $X$ and the eigenvalues of $\Sigma X \Sigma$, where $\Sigma$ is a diagonal matrix with all diagonal elements being positive real numbers?

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  • $\begingroup$ If $X$ is diagonal, it seems like the diagonal entries of $\Sigma X \Sigma$ can be (fill in the sentence...) $\endgroup$ – kimchi lover Jul 30 '17 at 21:45
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    $\begingroup$ is there any connection between the coefficients of $\Sigma^2$ and 1? $\endgroup$ – fonfonx Jul 30 '17 at 21:51
  • $\begingroup$ @fonfonx Thanks for comment. $Sigma$ is a general diagonal matrix. Only restriction is that all its elements are positive. $\endgroup$ – Ralph B. Jul 31 '17 at 12:04
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    $\begingroup$ @RalphB. this is why I think there is no connection between the eigenvalues of $X$ and the ones of $\Sigma X \Sigma$: for exemple take for $X$ the identity matrix: its eigenvalues are $1$. The eigenvalues of $\Sigma X \Sigma = \Sigma^2$ are the squares of the coefficients of $\Sigma$, which can take all positive values $\endgroup$ – fonfonx Jul 31 '17 at 12:48
  • $\begingroup$ @fonfonx Got it! Many thanks! $\endgroup$ – Ralph B. Jul 31 '17 at 14:43

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