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###

Singular Eigenproblems

In this section, we give a brief discussion of singular
matrix pairs **(***A*,*B*).

If the determinant of
is zero for all values of
(or the determinant of
is zero for all
),
the pair **(***A*,*B*) is said to be **singular**.
The eigenvalue problem of a singular pair is much more complicated
than for a regular pair.

Consider for example the singular pair

which has one finite eigenvalue 1 and one indeterminate eigenvalue 0/0.
To see that neither eigenvalue is well determined by the data,
consider the slightly different problem

where the
are tiny nonzero numbers. Then it is easy to see
that **(***A*',*B*') is regular with eigenvalues
and
.
Given *any* two complex numbers
and
,
we can find arbitrarily tiny
such that
and
are the eigenvalues of **(***A*',*B*').
Since, in principle, roundoff could change **(***A*,*B*) to **(***A*',*B*'), we
cannot hope to compute accurate or even meaningful eigenvalues of
singular problems, without further information.

It is possible for a pair **(***A*,*B*) in Schur form to be very close to singular,
and so have very sensitive eigenvalues, even if no diagonal entries of
*A* or *B* are small.
It suffices, for example, for *A* and *B* to nearly have a common null space
(though this condition is not necessary).
For example, consider the 16-by-16 matrices

Changing the **(***n*,1) entries of *A*'' and *B*'' to **10**^{-16}, respectively,
makes both *A* and *B* singular, with a common null vector.
Then, using a technique analogous to the one applied to **(***A*,*B*) above,
we can show that there is a
perturbation of *A*'' and *B*'' of norm
,
for *any* ,
that makes the 16 perturbed eigenvalues
have *any* arbitrary 16 complex values.
A complete understanding of the structure of a singular eigenproblem
**(***A*,*B*) requires a study of its *Kronecker canonical form*,
a generalization of the *Jordan canonical form*.
In addition to Jordan blocks for finite and infinite eigenvalues,
the Kronecker form can contain ``singular blocks'', which occur only
if
for all
(or if *A* and *B*
are nonsquare).
See [53,93,105,97,29] for
more details. Other numerical software, called GUPTRI,
is available for
computing a generalization of the
Schur canonical form for singular eigenproblems
[30,31].

The error bounds discussed in this guide hold for regular pairs only
(they become unbounded, or otherwise provide no information, when
**(***A*,*B*) is close to singular). If a (nearly) singular pencil is reported
by the software discussed in this guide, then a further study of the matrix
pencil should be conducted, in order to determine whether meaningful results
have been computed.

** Next:** Error Bounds for the
** Up:** Further Details: Error Bounds
** Previous:** Computing s_{i}, , and
** Contents**
** Index**
*Susan Blackford*

*1999-10-01*