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There is a similar question to mine posted here.

I have a matrix $L$ which is the graph Laplacian of a strongly connected normal graph. Therefore, $L$ is normal, has a simple eigenvalue at zero, and all its other eigenvalues have positive real parts.

Is there a way to express the eigenvalues of the product $DL$, (with $D>0$ a diagonal matrix) in terms of those of $L$ and $D$?

I found the following hints so far:

  • the spectra of $DL$ and $D^{1/2}LD^{1/2}$ are identical
  • $L$ and $D^{1/2}LD^{1/2}$ are congruent, and by Sylvester's law of inertia (extended to normal matrices), they have the same number of positive real part, negative real part and zero eigenvalues
  • The accepted answer to a similar question cites a paper on Horn's inequalities, but the paper is very advanced, and it is not clear how (and whether) its results apply to my question. Please note that $L$ is not necessarily symmetric in my case.
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