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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):

Constraints on an asymmetric $B$ would be fantastic, but any input is appreciated!

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Bobson Dugnutt
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  • $A$ is similar to $\sqrt{S}B\sqrt{S}$, which is congruent to $B$. When $B$ is symmetric, it has the same signature as $\sqrt{S}B\sqrt{S}$ (Sylvester's law of inertia) and hence eigenvalues of $A$ and $B$ have identical signs (because $A,\sqrt{S}B\sqrt{S}$ have the same spectra). – user1551 Dec 20 '16 at 10:39

1 Answers1

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Here is a partial answer, which leads to sufficient conditions. It uses a classic result of Bendixon (1902). Let $H(A) = (A+A^T)/2$ be the symmetric part of $A$. Then $$ \min_i \lambda_i\left( H(A) \right) \leq Re[\lambda_i(A)] \leq \max_i \lambda_i\left( H(A) \right). $$ As such, if we require that $Re[\lambda_i(A)] < 0$ it is sufficient that the eigenvalues of $H(A)$ are all negative. Equivalently, we may ask ourselves if we can find a positive definite matrix $Q$ such that $A + A^T + Q = 0$, i.e. $$ SB + B^TS + Q = 0. $$ This relation is called Lyapunov's equation. For a given $B$, Lyapunov's equation is known to have a unique positive definite solution $S$ for any positive definite $Q$ if and only if $Re[\lambda_i(B)] < 0$ for all $i$. In your case we also require that $S$ should be diagonal. There is a large amount of literature on diagonal solutions to Lyapunov's equation, but perhaps the most complete solution to the problem was given by Barker, Berman and Plemmons (1978): There is a diagonal solution $S$ if and only if $B X$ has at least one negative diagonal element for all non-zero positive semi-definite matrices $X$.

ekkilop
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