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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
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Susan Blackford
2000-11-20