We assume that and
are general
by
matrices.
We call
a matrix pencil, or pencil for short.
The most common case, and the one we will deal with first,
is the regular case, which occurs when
and
are square
and the characteristic polynomial
is not zero for
all
.
This is equivalent to assuming that there are
eigenvalues
(finite or infinite) and that they are
continuous functions of
and
, i.e., that small changes
in
and
cause small changes in the eigenvalues
(this requires an appropriate definition for the case of
infinite eigenvalues).
We will deal with the singular case at the end of this section.