The polynomial
is the
characteristic polynomial of
.
The degree of
is at most
.
The roots of
are called the finite eigenvalues of
.
If the degree of
is
, we say that
has
infinite eigenvalues too.
For example,
If is a finite eigenvalue,
a nonzero vector
satisfying
is a (right) eigenvector
for the eigenvalue
.
A nonzero vector
satisfying
is a left eigenvector.
If is an eigenvalue, nonzero vectors
and
satisfying
and
are called right and left eigenvectors, respectively.
An by
pencil
need not have
independent eigenvectors.
The simplest example is
, which is defined
in equation (2.3) and discussed in §2.5.1.
The fact that
independent eigenvectors may not exist
(though there is at least one for each distinct eigenvalue) will
necessarily complicate both theory and algorithms for the GNHEP.
Since the eigenvalues may be complex or infinite, there is no fixed way to order them.
Nonetheless, it is convenient to number them as
,
with corresponding right eigenvectors
and left eigenvectors
(if they exist).