Let us start by considering the generalized
eigenvalue problem
, where

The eigenvalues of are the pairs , ) and , ) with the associated eigenvectors and , respectively. If is nonzero, then is a finite eigenvalue. Otherwise, if is zero, then is an eigenvalue of the matrix pair . But what happens if, for example, ? Then is zero for all , which means that we have a singular eigenvalue problem. In this case we have ; i.e., and have a common null space. We say that is an eigenvector for an indeterminate eigenvalue . Note that the common null space is a sufficient but not necessary condition to have a singular eigenvalue problem.

The most common generalized eigenvalue problems
are *regular*;
i.e., and are square matrices and
the characteristic polynomial
is only vanishing for a finite number of values, where
denotes the degree of the polynomial.
The corresponding is called a regular matrix pencil.
The eigenvalues of a regular pencil are points in
the extended complex plane
.
The eigenvalues are defined as the zeros
of and additional eigenvalues.

An alternative representation of a matrix pencil
is the *cross product form*: the set of matrices
where
.
The mapping
shows the
relation between the eigenvalues of
and
.
For example, zero and infinite eigenvalues are represented as
and , respectively,
and can be treated as any other points in .

If
(and
)
is identically zero for all (and pairs (, )),
then is called *singular* and is a
singular matrix pair.
Singularity of is signaled by some
.
In the presence of roundoff, and
may be very small.
In these situations, the eigenvalue problem is very ill-conditioned, and
some of the other computed nonzero values of and
may be indeterminate.
Such problems are further discussed and illustrated by examples
in §8.7.4.
Moreover, rectangular matrix pairs are singular and the corresponding
is a singular pencil.