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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.

A nonzero vector satisfying
is a *(right) eigenvector*
for the eigenvalue .
Since
,
left and right eigenvectors are identical.

All eigenvalues of a Hermitian matrix 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 eigenvalue has an eigenvector .
We may assume without loss of generality that .
Each is real if is real.
Though the may not be unique (e.g., any vector is an eigenvector of
the identity matrix), they may be chosen to all be orthogonal to one another:
if .
When an eigenvalue is distinct from all the other eigenvalues,
its eigenvector is unique (up to multiplication by scalars).

** Next:** Invariant Subspaces
** Up:** Hermitian Eigenproblems J.
** Previous:** Hermitian Eigenproblems J.
** Contents**
** Index**
Susan Blackford
2000-11-20