Let $A\in \mathbb{R}^{n\times n}$. Moreover, assume $D$ is an $n \times n$ diagonal matrix with positive diagonals. What is the relation between the eigenvalues of $A$ and eigenvalues of $B:=DAD$? In other words, how do the eigenvalues of the matrix $A$ change when it is pre and post multiplied by a diagonal matrix? Do $A$ and $B$ have the same inertia? We can assume that $A$ is diagonalizable if necessary.

Any comment/response is greatly appreciated.

  • $\begingroup$ $A$ and $B$ are congruent (since $D=D^T$) so they do have the same inertia $\endgroup$ – Omnomnomnom Jan 12 at 20:49
  • $\begingroup$ Thanks for the comment. How about the location of eigenvalues. Is there any result on how the eigenvalues of $A$ would move? $\endgroup$ – Arthur Jan 12 at 20:55
  • $\begingroup$ Is $A$ symmetric? $\endgroup$ – Omnomnomnom Jan 12 at 21:09
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    $\begingroup$ Bhatia's Matrix Analysis is usually a good reference for these purposes, but I'm having trouble finding inequalities that would apply to your specific situation. If you were content to consider the singular values rather than eigenvalues, then we could note that $DAD$ is similar to $D^2 A$, and that $$ \prod_{j=1}^k \sigma_j(D^2A) \leq \prod_{j=1}^k \sigma_j(D^2) \prod_{j=1}^k \sigma_j(A) $$ The singular values of $D^2$, since $D^2$ is diagonal with positive diagonals, are just those values on the diagonal. $\endgroup$ – Omnomnomnom Jan 13 at 16:51
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    $\begingroup$ You can find more inequalities along these lines in section III.4 of Bhatia's text (Lidskii's theorems) $\endgroup$ – Omnomnomnom Jan 13 at 16:53

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