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##

Related Eigenproblems

- Consider the GNHEP
,
where and are square and is nonsingular. The
matrix has the same eigenvalues and right eigenvectors as
. If is a left eigenvector, i.e.,
,
is a left eigenvector of . Analogous statements are
true about . If is a factorization of
(from Gaussian elimination,
QR decomposition, or anything else),
has the same
eigenvalues as , right eigenvector , and left eigenvector .
If is well-conditioned, or , , or
can be accurately computed, this is an effective way to solve
. If is ill-conditioned, it is preferable to
treat it as the GNHEP,
see §2.6.

- Let
be a monic
polynomial. Define the by
*companion matrix of * as

where all entries not explicitly shown are 0. Then the eigenvalues
of are the roots of
, and the right
eigenvectors
are
. is
not diagonalizable if has multiple roots.
A reliable, but not optimally efficient, algorithm for finding roots
of a polynomial is to find all the eigenvalues of .

- Let
be a monic matrix polynomial,
where each is an by matrix.
An eigenpair
of
satisfies
.
Define the by
*block companion matrix of * as

where all entries are by blocks and all entries not explicitly shown
are 0.
Then 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:** Non-Hermitian Eigenproblems J. Demmel
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Susan Blackford
2000-11-20