TITLE: Perturbation theory for the generalized eigenvalue problem
QUESTION [9 upvotes]: 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).

REPLY [6 votes]: In addition to those already suggested by Federico, a good reference is Chapter 2 in Kato's book Perturbation Theory for Linear Operators, especially if you want to tackle the question from the higher perspective of functional analysis. Another reference, addressing the same problem from the alternative point of view of the calculus of variations (with an eye on computations), is I. Babuška and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis (Part 1), Vol. II, ed. by P.G. Ciarlet and J.L. Lions, pp. 641-787 (1991).