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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].
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Up: Non-Hermitian Eigenproblems J. Demmel
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Susan Blackford
2000-11-20