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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
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Susan Blackford
2000-11-20