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##

Related Eigenproblems

Since the HEP is one of the best understood
eigenproblems, it is helpful to recognize when other eigenproblems
can be converted to it.

- If is non-Hermitian, but
is
Hermitian for easily determined and ,
it may be advisable
to compute the eigenvalues and eigenvectors of .
One can convert these to eigenvalues and eigenvectors of via
and
.
For example, multiplying a skew-Hermitian matrix (i.e., )
by the constant makes it Hermitian.
See §2.5
for further discussion.

- If for some rectangular matrix , then the eigenproblem
for is equivalent to the SVD of , discussed
in §2.4.
Suppose is by , so is by .
Generally speaking, if is about as small or smaller than
(, or just a little bigger), the eigenproblem for is
usually cheaper than the SVD of . But it may be less accurate to compute
the small eigenvalues of than the small singular values of .
See §2.4
for further discussion.

- If one has the generalized HEP
,
where and are Hermitian and is positive definite,
it can be converted to a Hermitian eigenproblem as follows.
First, factor , where is any nonsingular matrix (this is
typically done using Cholesky factorization). Then solve the HEP
for
. The eigenvalues of
and are identical, and if is an
eigenvector of , then
satisfies
.
See §2.3
for further discussion.

** Next:** Example
** Up:** Hermitian Eigenproblems J.
** Previous:** Specifying an Eigenproblem
** Contents**
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Susan Blackford
2000-11-20