# Let $X$ be positive definite, any connection between the eigenvalues of $X$ and the eigenvalues of $\Sigma X \Sigma$, $\Sigma$ is positive diagonal?

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?

• If $X$ is diagonal, it seems like the diagonal entries of $\Sigma X \Sigma$ can be (fill in the sentence...) – kimchi lover Jul 30 '17 at 21:45
• is there any connection between the coefficients of $\Sigma^2$ and 1? – fonfonx Jul 30 '17 at 21:51
• @fonfonx Thanks for comment. $Sigma$ is a general diagonal matrix. Only restriction is that all its elements are positive. – Ralph B. Jul 31 '17 at 12:04
• @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 – fonfonx Jul 31 '17 at 12:48
• @fonfonx Got it! Many thanks! – Ralph B. Jul 31 '17 at 14:43