** Next:** Equivalences.
** Up:** Singular Case
** Previous:** Eigenvalues and Eigenvectors.
** Contents**
** Index**

####

Reducing Subspaces.

A *pair of right and left reducing subspaces* and
of
satisfy and
for all , and furthermore
.
Also, the dimension of exceeds the dimension of by
, the dimension of the right null space of
over the field of all rational functions of
[119].
It is easy to express
in terms of the
Kronecker canonical form, described below.
There is still a correspondence between subsets of eigenvalues
and reducing subspaces (as there was a correspondence between
subsets of eigenvalue of invariant subspaces for single matrices), but
reducing subspaces are no longer spanned by eigenvectors, which are
no longer well defined.

Consider the singular pencil (2.6).
It has a nontrivial right reducing subspace spanned by the first two
columns of the 3 by 3 identity matrix and corresponding left reducing subspace
spanned by . Here is spanned by
.

Susan Blackford
2000-11-20