I'm trying to find a relation between the signs of the eigenvalues of a matrix $A=SB$ and the eigenvalues of $B$, when all three are invertible and $S$ and $B$ exclusively have real entries, and $S$ is a positive diagonal matrix.
Specifically, I demand that $A$'s eigenvalues satisfy $Re[\lambda_i]<0 \;\forall i$. In this case, what are the constraints on $B$, i.e., what can we say about it, its eigenvalues and their signs?
From simulations I know that there is, in general, no conservation of the signs of the eigenvalues. However it seems that when $B$ is symmetric, the number of eigenvalues with a specific sign is conserved, i.e., its signature is preserved. Is this true? I have looked at Sylvester's Law of Inertia (wiki, mathworld), but since I don't know anything about quadratic forms, I don't really know what to do with it. I got the impression that it might be of use through this MO answer.
From my looking around, these questions seem useful (although, I don't really get the theory behind the answers, so I'm unsure whether or not they apply to my problem):
- Eigenvalues of product of a matrix and a diagonal matrix (which mentions Horn's inequalities)
- Eigenvalues of product of a matrix with a diagonal matrix (funnily, they have the same name!)
- if eigenvalues are positive, is the matrix positive definite?
- Does non-symmetric positive definite matrix have positive eigenvalues?
- square root of symmetric matrix and transposition (which mentions Cholesky factorization)
Constraints on an asymmetric $B$ would be fantastic, but any input is appreciated!