I have:
- a symmetric Laplacian matrix $L$ with the usual zero eigenvalue and all other eigenvalues have positive real parts, and:
- a strictly diagonally dominant matrix $M$ with strictly positive diagonal terms and thus with all eigenvalues having positive real parts. Note that $M$ is not diagonal - there are (small) off-diagonal terms which are not symmetric in any way.
My question is, whether I can bound the eigenvalues of $LM$ in any way? More particularly, I am interested in proving that the eigenvalues of $LM$ are have non-negative real parts.
My research so far:
Numerical simulations seem to indicate that this is the case, but I am ideally looking for a proof or for some theoretical condition on $M$ that guarantees that $LM$ has no negative eigenvalues for any Laplacian $L$ which has the properties above.
$LM$ can be interpreted as the Laplacian matrix of a directed, signed graph (i.e., with some negative edge weights). However, it is known that such graphs may have negative eigenvalues, e.g., [1].
[1] https://www.sciencedirect.com/science/article/pii/S0024379517301301
Many thanks in advance for answers / guidance / helpful references.