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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*