Is there a standard reference for the perturbation theory of the generalized eigenvalue problem?
More specifically, I would like to get a systematic expansion for the problem
$(A_0 + \epsilon A_1)v = \lambda B v$
in terms of the solutions to
$A_0 v = \lambda B v$
where $A_0$, $A_1$ and $B$ are known, $n\times n$ Hermitian matrices, $\epsilon >0 $ is a "small" parameter, and $\lambda\in \mathbb{C}$ and $v\in\mathbb{C}^n$ are the unknowns. When $B$ is positive-definite, the usual (Rayleigh-Schroedinger) perturbation series can easily be generalized by using $B$ to define an alternative inner product. However, in my problem, $B$ is not positive-definite (although it is still nonsingular).
Best Answer
The standard book is Stewart, Sun, Matrix perturbation theory. It has a part devoted only to the generalized eigenvalue problem. You may want to check out some individual papers of Stewart and Sun as well.
There are some remarks in Golub, Van Loan, Matrix computations as well. I think they mention the fact that the Hermitian/Hermitian case is not as well-behaved as the Hermitian/definite case, confirming your suspicions.
In any case, here is a nifty trick that can help you get a first-order expansion, and that can perhaps be generalized. If the eigenvalue is simple, we know that the perturbed eigenvalue and eigenvector is an analytic function of the perturbation. Therefore, we may take derivatives in $$ Av=\lambda Bv $$ to get $$ (\Delta A)v+A\dot{v}=\dot{\lambda}Bv+\lambda (\Delta{B})v+\lambda B\dot{v}. $$ Now multiply by the left eigenvector $u^* $ to get $$ u^* (\Delta A)v = \dot{\lambda} u^* Bv+\lambda u^*(\Delta B)v. $$ Solve for $\dot{\lambda}$ and you get a first-order expression for the perturbed eigenvalue. (Yes, I know it sounds fishy, but I think everything can be made rigorous with a little work).