Let
be an eigenvalue of matrix
with
algebraic multiplicity
.
A vector
with
is called generalised eigenvector for eigenvalue
.
All generalised eigenvectors together with the zero vector form
a subspace of dimension
called generalised eigenspace
for eigenvalue
. This subspace is invariant
under the linear mapping
.
Since the identity matrix commutes with each matrix
we obtain for the image
of a vector
:
Consequently, we have
.
In order to find the dimension we bring matrix
to upper triangle form:
For
we have
thus, the generalised eigenvectors transform
according to
.
By the form of
it can easily be seen that
for
the unit vectors
belong to
the generalised eingenspace
of
but for von
they do not.
Consequently we have
.
(Authors: Burkhardt/Höllig/Hörner)
|
automatically generated
4/21/2005 |