Introduction

A generalized Hermitian eigenvalue problem (GHEP) is given by

$$Ax=\lambda B x\,,$$ (66)

where $A$ and $B$ are Hermitian, $A^{\ast} = A$, and $B^{\ast} = B$. We call the pair $\{A,B\}$ of matrices in (5.1) a matrix pencil. In this chapter we make the additional assumption that $A$ or $B$ or $\alpha A + \beta B$ for some scalars $\alpha$ and $\beta$ is positive definite, in which case we talk about a Hermitian definite pencil. This assumption is true for a wide class of practically important cases, and the theory is very closely related to the standard Hermitian eigenproblem, as expounded in Chapter 4. If no positive definite combination exists, we could as well regard $\{A,B\}$ as a general pencil and use the theory and algorithms described in Chapter 8.

The nonstandard case, where $\alpha A + \beta B$ is positive definite, may be reduced to the standard case when $B$ is positive definite, by noting that the pencil

$$(A-\theta(\alpha A + \beta B))x=0$$ (67)

has eigenvalues $\theta_i=\lambda_i/(\beta+\alpha \lambda_i)$ and the same eigenvectors as the original pencil (5.1). One may apply any algorithm applicable for positive definite $B$ to this modified pencil and recover the $\lambda_i$ from the $\theta_i$.

The GHEP (5.1) has $n$ real eigenvalues $\lambda_i$, which we may order increasingly so that $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. Several eigenvalues may coincide, as in the standard case, except that some eigenvalues may be infinite. If the matrices $A$ and $B$ are positive definite, $\lambda_1>0$, but if $A$ is positive semidefinite we can only say that $\lambda_1\geq 0$.

When the pencil $\{A,B\}$ is Hermitian definite, it is possible to find $n$ mutually $B$-orthogonal eigenvectors, $x_i,\,i=1,\dots,n$, so that

$$A=X \Lambda X^{\ast},\quad B=X X^{\ast}, \pagebreak$$ (68)

where $\Lambda=\diag(\lambda_i)$ and $X=[x_1,x_2,\dots,x_n]$. The eigenvectors $x_i$ are not unique, but there is a unique reducing subspace for each different eigenvalue. For a Hermitian definite pencil, the reducing subspace is of the same dimension as the multiplicity of the eigenvalue. It is important to keep in mind that when a couple of eigenvalues coincide, their eigenvectors lose their individuality: there is no way of saying that one set of vectors comprises the eigenvectors of a multiple eigenvalue.

There are some cases where the matrix $B$ is singular (positive semidefinite). We still have a Hermitian definite pencil provided that $Ax\neq 0$ for all vectors $x$ in the null space of $B$. The pencil (5.1) then has a $p$-fold infinite eigenvalue with the null space of $B$ as its reducing subspace. In most practical cases, we are not interested in computing these infinite eigenvalues, since they correspond to constraints, symmetries or rigid body modes of the physical application that gave rise to the eigenvalue computation.

Eigenvalues $\{\lambda_i\}$ of $A - \lambda B$ may be well-conditioned or ill-conditioned. If $B=I$ (or close to it), the eigenvalues are well-conditioned (close to it) as for the Hermitian eigenproblem. But if $\vert x^{\ast}_i B x_i\vert$ is small, where $x_i$ is a unit eigenvector of $\lambda_i$, $\lambda_i$ can be very ill-conditioned. We give a more detailed account of error assessment for the computed eigenvalues and eigenvectors in §5.7.