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##

Eigenvalues and Eigenvectors

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,

has characteristic polynomial
,
and so has eigenvalues , , and .
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).

** Next:** Deflating Subspaces
** Up:** Generalized Non-Hermitian Eigenproblems
** Previous:** Generalized Non-Hermitian Eigenproblems
** Contents**
** Index**
Susan Blackford
2000-11-20