# Perturbation Theory for the Definite Generalized Eigenvalue Problem

I have become confused on some basic questions concerning the perturbation theory of the generalized eigenvalue problem. I am likely missing something simple, but I have a couple of questions about the literature on this subject.

Consider the generalized eigenvalue problem $$Ax = \lambda Bx$$ for $$n$$-square Hermitian matrices $$A$$ and $$B$$. The Crawford number is defined to be

$$c(A,B) := \inf_{x\in \mathbb{C}^n, \|x\|_2=1} |x^*(A+iB)x|.$$

Assume $$c(A,B) > 0$$, which ensures that there exists $$Q$$ such that $$Q^*AQ$$ and $$Q^*BQ$$ are simultaneous diagonalized, the generalized eigenvalues $$\lambda$$ being the ratio of these diagonal entries. For each generalized eigenvalue $$\lambda_i$$ with eigenvector $$q_i$$, the associated angle is defined to be the angle between a fixed ray and the complex number $$\theta_i = q_i^*Aq_i + iq_i^*Bq_i$$.

I am interested in perturbed versions of this problem, namely $$(A+E)x = \lambda (B+F)x$$ where $$\|E\|_{\rm op}^2 + \|F\|_{\rm op}^2 =: \epsilon < c(A,B)$$. Here, $$\|E\|_{\rm op}$$ is the spectral (operator 2-) norm. Stewart (1979, Theorem 3.2) derives the result

$$|\theta_i - \tilde{\theta}_i|\le \arcsin\left( \frac{\epsilon}{c(A,B)} \right) =: \mathrm{RHS}_1.$$

Here, $$\tilde{\theta}_i$$ are the angles associated with the eigenvalues of the perturbed problem. In Golub and Van Loan (2012, Theorem 8.7.3), they cite Stewart (1979) as saying

$$| \arctan(\lambda_i) - \arctan(\tilde{\lambda_i})| \le \arctan\left( \frac{\epsilon}{c(A,B)} \right) =: \mathrm{RHS}_3.$$

I believe it's the case that $$| \arctan(\lambda_i) - \arctan(\tilde{\lambda_i})| = |\theta_i - \tilde{\theta}_i|$$, which would make the result of Golub and Van Loan (2012) a strict strengthening of the result from Stewart (1979) since $$\mathrm{RHS}_3 \le \mathrm{RHS}_1$$.

In deriving his result, Stewart reduces his problem to an elementary question of geometry. By my reading, this problem is as follows: given a circle $$C_1$$ with radius $$c(A,B)$$ and another circle $$C_2$$ centered at a point on $$C_1$$ with radius $$\epsilon$$, what is the angle of the arc between the center of $$C_2$$ and an intersection point of $$C_1$$ and $$C_2$$. Stewart states the solution of this problem is $$\mathrm{RHS}_1$$, but my derivation is that the solution to this problem is actually $$\mathrm{RHS}_2 := 2\arcsin(\epsilon/(2c(A,B))$$.

For the relevent region $$0 < \epsilon < c(A,B)$$, $$\mathrm{RHS}_1 \ge \mathrm{RHS}_2 \ge \mathrm{RHS}_3$$. Here are my two questions:

• Am I correct that Stewart (1979)'s analysis actually leads to the stronger bound $$\mathrm{RHS}_2$$ for the left-hand side?
• Is the bound $$\mathrm{RHS}_3$$ correct, and if so, where is a correct proof? Is it contained in Stewart (1979)?

Any references would be great. I am aware the monograph by Stewart and Sun contains a discussion of this problem, but I do not have access to a copy of this book right now.

My purported improved bound $$\mathrm{RHS}_2$$ is incorrect (at least for the reasons I believed it) because I misinterpreted the geometry problem. The problem is actually
Given a circle $$C_1$$ with radius $$c(A,B)$$ and another circle $$C_2$$ centered at a point on $$C_1$$ with radius $$\epsilon$$, what is the maximum angle between the center of $$C_2$$, the center of $$C_1$$, and a point on $$C_2$$.
In this case, the maximizer is when the triangle described by this angle is a right triangle with hypotenuse $$c(A,B)$$ and opposite leg $$\epsilon$$. This gives the bound $$\mathrm{RHS}_1 = \arcsin(\epsilon/c(A,B))$$ as claimed by Stewart.
Now having gotten my hands on Matrix Perturbation Theory and done more reading, I have not seen the $$\mathrm{RHS}_3$$ bound anywhere in the literature except for Golub and Van Loan, which cite Stewart 1979 for a proof which (unless I'm really missing something) it does not contain.
I am now reasonably convinced $$\mathrm{RHS}_1$$ is the best possible bound for this problem of this flavor only depending on $$\epsilon$$ and $$c(A,B)$$. (There are certainly stronger bounds, such as those by Sun (1982) and Mathias and Li (2004), but they use different parameters than $$\epsilon$$ and $$c(A,B)$$.)