Singular pencils may or may not have eigenvalues.
Indeed, the generic case corresponds to a singular
pencil that has no eigenvalues.
Below we illustrate this with two 3 by 3 examples:
If is upper triangular and a zero element appears on the
diagonal, then the pencil is singular.
We see that both examples have this property
(the
entries of
and
are zero as well as
for
and
).
This situation will appear if we apply the
QZ algorithm to a square singular pencil in infinite precision.
Such a pair (
,
) = (0, 0) is called an indeterminate
eigenvalue 0/0.
In the presence of roundoff, the QZ algorithm may fail to
detect and isolate the singularity due to the ill-conditioning
of the problem as illustrated below.
The eigenvalue problem for a singular pair is much more
complicated than for a regular pair.
Consider, for example, the singular pair
A well-known class of singular pencils is the class of matrix
pairs with intersecting null spaces.
Let belong to the intersection of the null spaces of
and
, i.e.,
.
Then for any
, we have
,
implying that the pair
is singular.
By inspection we see that
and
above have a common one-dimensional
column (and row) null space spanned by
.
The dimensions of the intersecting column and row null spaces, respectively,
are the number of
and
blocks, respectively, in the
pencils' KCF.
Notice that the intersection of null spaces of
and
is a sufficient
but not a necessary condition for a pencil to be singular, as
illustrated with
in the first set of examples.
It is possible for a pair in Schur form to be very close to
singular, and so to have very sensitive eigenvalues, even if no diagonal
entries of
or
are small. It suffices for
and
to
nearly have a common null space.
For example, consider the
matrices
With these examples in mind we are ready to introduce the GUPTRI form and the regularization method used to compute meaningful and reliable information.