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Introduction
In this chapter we treat the standard Hermitian, or most often real
symmetric, eigenvalue problem (HEP),

(16) 
where .
It is the most commonly occurring algebraic eigenvalue problem, for which
we have the most reliable theory and the most powerful algorithms available.
A summary of the basic theory about
the Hermitian eigenvalue problem (4.1)
is given in §2.2.
It is known that the problem (4.1) has real eigenvalues
, which we may order increasingly so that
. Note that several
eigenvalues may coincide. In many applications the matrix is
positive definite, (or positive semidefinite,
).
Even in the cases
where positive definiteness can be used to advantage,
we choose to treat Hermitian with a general
distribution of eigenvalues in this chapter.
When is Hermitian, it is always possible to find mutually
orthogonal eigenvectors,
, so that

(17) 
where
,
, and
. The eigenvectors are not unique; what is unique is the
invariant subspace for each different eigenvalue.
For a Hermitian matrix , the invariant subspace is of the same dimension
as the multiplicity of the eigenvalue. It is important to keep in mind that
when certain eigenvalues coincide, their eigenvectors lose
their individuality: there is no way of saying that one set of vectors
are the eigenvectors of a multiple eigenvalue.
Two different algorithms, and even two different runs with the same algorithm,
will give different representative eigenvectors in the invariant subspace.
On the other hand, a user is entitled to demand that an algorithm produce
eigenvector approximations that are orthogonal to each other,
even if the matrix has multiple eigenvalues.
The eigenvalues of are always wellconditioned.
If a perturbation is added to the matrix ,
then each of the eigenvalues is perturbed at most as far as
the spectral norm ,

(18) 
There are several stronger results available, like the
WielandtHoffman inequality that says that the sum of squares
of the differences between the eigenvalues is majorized
by the sum of squares of the elements of ; see §4.8 and
references therein.
In many cases, one is interested in eigenvalues that are small compared
to the norm , and for such eigenvalues the inequality (4.3)
is not very satisfactory, since it allows large
relative perturbations for such small eigenvalues. It is possible
to develop perturbation bounds for relative perturbations under certain
additional assumptions.
There is no result as simple as (4.3) available for
bounding the perturbation of eigenvectors: one has to add an
assumption of separation between the eigenvalues.
Let us cite the
venerable theorem of Davis and Kahan from [101].
Let us assume that is
an approximation of the eigenpair
of ,
where is normalized so that .
The ``best'' corresponding to
is the Rayleigh quotient
,
so we assume that has this value.
Suppose that is closer
to than any other eigenvalues of , and
let be the gap between
and any other eigenvalue:
.
Define the residual vector as ; then we have

(19) 
Under the same assumptions, we have the improved bound on the eigenvalue
approximation,
_i{r_2,r_2^2}.
The first alternative in this minimum is equivalent to the previous
bound (4.3), since is an eigenvalue of the perturbed
matrix with a rankone perturbation of
norm
(it is not necessary for to be
Hermitian in the bound (4.3)).
The reader is referred to §4.8 for further details.
Subsections
Next: Overview of Available Algorithms.
Up: Hermitian Eigenvalue Problems
Previous: Hermitian Eigenvalue Problems
Contents
Index
Susan Blackford
20001120