Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix and $D = \text{diag} (d_1, d_2, ... , d_n)$ be a positive diagonal matrix. We know that eigenvalues of A are $\lambda_1 \leq \lambda_2 \leq ... \leq \lambda_n$. What will be eigenvalues of $DAD$? (Obviously they're all positive.)
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2$\begingroup$ You don't mean $DAD^{-1}$, right? Similar matrices have the same eigenvalues (math.stackexchange.com/questions/8339/…) $\endgroup$ – parsiad Aug 8 '15 at 3:21
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$\begingroup$ Absolutly, I meant $DAD$. You're right about similar matrices though. $\endgroup$ – questioner Aug 8 '15 at 4:09
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2$\begingroup$ It's impossible to give an exact result without more information about $A$. However, we might be able to get some useful inequalities. $\endgroup$ – Omnomnomnom Aug 8 '15 at 4:28
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$\begingroup$ Assume $A$ is laplacian of an undirected connected graph. (I know $A$ is positive semi-definite in this case). What inequalities? (beside the most obvious ones) $\endgroup$ – questioner Aug 8 '15 at 4:32
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$\begingroup$ I'd have to look through Bhatia's matrix analysis to see if something applies; I can't think of anything relevant off the top of my head. $\endgroup$ – Omnomnomnom Aug 8 '15 at 12:37
