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Eigenvalues and Eigenvectors
The polynomial
is called the
characteristic polynomial
of . The roots of
are called the eigenvalues of .
Since the degree of is , it has roots, and so
has eigenvalues. Eigenvalues of a real matrix may be real or appear in
complex pairs.
A nonzero vector satisfying
is a (right) eigenvector
for the eigenvalue .
A nonzero vector satisfying
is a left eigenvector
for the eigenvalue .
An by matrix need not have independent eigenvectors.
The simplest example is

(3) 
whose eigenvalues are
.
When
, there are two right eigenvectors,
. As approaches 0, the
two eigenvectors approach one another.
When , both eigenvalues equal 0,
and there is a single independent right eigenvector parallel to .
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 NHEP.
Since the eigenvalues may be complex, 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: Invariant Subspaces
Up: NonHermitian Eigenproblems J. Demmel
Previous: NonHermitian Eigenproblems J. Demmel
Contents
Index
Susan Blackford
20001120