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Suppose there is a Hermitian matrix $S$ with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_K$. There is a diagonal matrix $D$ whose entries on the main diagonal are positive. What are the eigenvalues of $DSD$?

The answer is obvious when all entries of $D$ are equal, but how about beyond this case?

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First of all, note that $DSD$ and $SD^2$ have the same eigenvalues, so you are asking how the eigenvalues $\lambda_n$ of a Hermitian matrix $S$ change when it is multiplied by a diagonal matrix $D^2$ with positive diagonal elements $\delta_n^2$. As you can read in this MSE posting, the eigenvalues $\mu_n$ of $SD^2$ are related to $\lambda_n$ and $\delta_n$ by a variation on Horn's inequalities. If you order the eigenvalues as $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_K$, $\delta_1\geq\delta_2\geq\cdots\geq \delta_K$, $\mu_1\geq\mu_2\geq\cdots\geq \mu_K$, then $$\mu_{i+j-1}\leq \lambda_i\delta_j^2,\;\;\mu_{i+j-K}\geq\lambda_i\delta_j^2.$$

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  • $\begingroup$ This helps me a lot. Thank you very much! $\endgroup$
    – Tina
    Mar 29 '18 at 7:16

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