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##

Eigenvalues and Eigenvectors

The polynomial
is called the
*characteristic polynomial*
of . The roots of
are *eigenvalues* of .
Since the degree of is , it has roots, and so
has eigenvalues.

A nonzero vector satisfying
is a
*(right) eigenvector*
for the eigenvalue . The eigenpair also satisfies
, so we can also call a *left eigenvector*.

All eigenvalues of the definite pencil are real.
This lets us write
them in sorted order
.
If all , then is called
*positive definite*,
and if all
, then is called
*positive semidefinite*.
*Negative definite* and *negative semidefinite* are defined
analogously. If there are both positive and negative eigenvalues,
is called *indefinite*.

Each is real if and are real.
Though the may not be unique,
they may be chosen to all be * orthogonal* to one another:
if . This is also called *orthogonality
with respect
to the inner product induced by the Hermitian positive definite matrix .*
When an eigenvalue is distinct from all the other eigenvalues,
its eigenvector is unique (up to multiplication by scalars).

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