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A generalized Hermitian eigenvalue problem (GHEP) is given by

(66) 
where and are Hermitian, , and .
We call the pair of matrices in (5.1)
a matrix pencil.
In this chapter we make the additional assumption that
or or
for some scalars and
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
as a general pencil and use the theory and algorithms described
in Chapter 8.
The nonstandard case, where
is positive definite,
may be reduced to the standard case when is positive definite,
by noting that the pencil

(67) 
has eigenvalues
and the same eigenvectors
as the original pencil (5.1).
One may apply any algorithm applicable
for positive definite to this modified pencil and recover the
from the .
The GHEP (5.1) has real eigenvalues
, which we may order increasingly so that
. Several
eigenvalues may coincide, as in the standard case,
except that some eigenvalues
may be infinite. If the matrices and are
positive definite, , but if is positive semidefinite
we can only say that
.
When the pencil is Hermitian definite,
it is possible to find mutually
orthogonal eigenvectors,
, so that

(68) 
where
and
.
The eigenvectors 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 is singular (positive
semidefinite). We still have a Hermitian definite pencil provided
that for all vectors in the null space of . The
pencil (5.1) then has a fold infinite eigenvalue with
the null space of 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 of may be wellconditioned
or illconditioned. If (or close to it), the eigenvalues
are wellconditioned (close to it) as for the Hermitian eigenproblem.
But if
is small, where is a unit
eigenvector of , can be very
illconditioned. We give a more detailed account of error assessment
for the computed eigenvalues and eigenvectors in §5.7.
Subsections
Next: Overview of Available Algorithms.
Up: Generalized Hermitian Eigenvalue Problems
Previous: Generalized Hermitian Eigenvalue Problems
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Susan Blackford
20001120