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##

Related Eigenproblems

- If is nonsingular, then the NHEP
has the same eigenvalues and
corresponding right eigenvectors as
.
Similarly,
has the same eigenvalues
as
and right eigenvectors
.
If is nonsingular,
has the
same right eigenvectors as
, and its eigenvalues are
reciprocals
.
Finally, if is nonsingular,
has reciprocal eigenvalues
and right eigenvectors .
Analogous statements can be made about left eigenvectors.

- More generally, suppose
has eigenvalues
and corresponding right eigenvectors .
Let , , , and
be scalars such that
.
Then
has the same eigenvectors as
and
eigenvalues
.
If one or both of
and
are nonsingular, then the method in item 1 above can be applied.

- Let
be
an -by- matrix polynomial, where
is not
identically zero.
An eigenpair
of satisfies
.
Define the by
*block companion pencil of* as

where all entries are by blocks and all entries not explicitly shown
are 0.
Then
is a regular generalized eigenvalue problem,
and the eigenvalues of are the eigenvalues of .
Note that there are eigenvalues.
If
is an eigenpair of , then
is a
right eigenvector of [194].

** Next:** Example
** Up:** Generalized Non-Hermitian Eigenproblems
** Previous:** Specifying an Eigenproblem
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Susan Blackford
2000-11-20