# Perturbation theory for the generalized eigenvalue problem

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).

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Well, if $B$ is invertible, why don't you multiply both sides by $B^{-1}$? Then at least your system is a little simpler-looking. –  MTS Jun 4 '12 at 3:55
Various nice properties (e.g., orthogonality) of the eigenvectors follow easily from the Hermiticity of $A$ and $B$, and I was hoping that the perturbation theory would be better-behaved in this form. The general theory of perturbations of eigenvalues (of a possibly non-normal matrix, which $B^{-1}A$ may be) seemed a little scary, but maybe it will turn out to be ok, and I may end up trying that. The case of positive-definite $B$ was an almost trivial generalization of the usual perturbation theory for a Hermitian matrix with all the usual niceties, so I was hoping to have something similar. –  user142 Jun 4 '12 at 4:07
For one example of what can go wrong if $B$ is not positive-definite, consider the case $A = \pmatrix{1 & 1\cr 1 & 1\cr}$, $B = \pmatrix{1 & 0\cr 0 & -1\cr}$, where $Av = \lambda B v$ is "missing" a generalized eigenvector: the Jordan form of $B^{-1} A$ is $\pmatrix{0 & 1\cr 0 & 0\cr}$. –  Robert Israel Jun 4 '12 at 6:06
Thanks, Robert, that was an eye-opener. –  user142 Jun 4 '12 at 6:37

## 2 Answers

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).

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Note to others who are wondering whether to buy Steward and Sun: The focus is on establishing perturbation bounds, not on expansions. The first order expansion given here is mentioned in passing, but nothing on second order. This is clearly mentioned in the introduction... –  Janus Aug 13 at 6:24

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).

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