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Related Eigenproblems
- If
and
are Hermitian,
is not positive definite, but
is positive definite for some choice of real numbers
and
, one can solve the generalized Hermitian
eigenproblem
instead.
Let
;
then the eigenvectors of
and
are identical.
The eigenvalues
of
and
the eigenvalues
of
are related by
.
- If
and
are non-Hermitian, but
and
are Hermitian, with
positive definite,
for easily determined
,
and nonsingular
and
,
then one can compute the eigenvalues
and eigenvectors
of
.
One can convert these to eigenvalues
and eigenvectors
of
via
and
.
For example, if
is Hermitian positive definite but
is skew-Hermitian
(i.e.,
), then
is Hermitian, so we may choose
,
, and
.
See §2.5
for further discussion.
- If one has the GHEP
,
where
and
are Hermitian and
is positive definite, then
it can be converted to a HEP 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
.
Indeed, this is a standard algorithm for
.
- If
and
are positive definite with
and
for some rectangular matrices
and
,
then the eigenproblem for
is equivalent to the quotient singular value decomposition (QSVD)
of
and
, discussed in §2.4.
The state of algorithms is such that it is probably better to try solving
the eigenproblem for
than computing the QSVD of
and
.
Next: Example
Up: Generalized Hermitian Eigenproblems
Previous: Specifying an Eigenproblem
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Susan Blackford
2000-11-20