Next: Reducing Subspaces.
Up: Singular Case
Previous: Singular Case
  Contents
  Index
Eigenvalues and Eigenvectors.
The singular case of
corresponds to either
is square and singular for all values of
, or
is rectangular.
Both cases arise in practice, and are significantly more challenging than
the regular case. We outline the theory here and leave details to
§8.7.
Consider
![\begin{displaymath}
A - \lambda B = \bmat{cc} 1 & 0 \\ 0 & 0 \emat -
\lambda \bmat{cc} 1 & 0 \\ 0 & 0 \emat.
\end{displaymath}](img591.png) |
(4) |
Then
for all
, so
is singular. For any
,
for
. But rather than calling all
eigenvalues, we
only call
an eigenvalue of this pencil, because
for
, and the rank of
is 0, which is lower than
the rank of
for any other value of
. In general,
if
has a lower rank than the rank of
for
almost all other values of
, then
is an eigenvalue.
Eigenvalues are discontinuous functions of the matrix entries when the
pencil is singular, which is one reason we have to be careful about
definitions. This discontinuity is further discussed below.
Eigenvectors are also no longer so simply defined.
For example, consider
![\begin{displaymath}
A - \lambda B = \bmat{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \emat
-\lambda \bmat{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \emat.
\end{displaymath}](img600.png) |
(5) |
Then
is an eigenvalue but
for any
for any value of
.
Instead we consider reducing subspaces, as defined below.
![\begin{displaymath}
% latex2html id marker 1839For example, change $A_{22}$\ t...
...ussed in more detail in
section~\ref{sec_chap2_GnHepCondSing}.
\end{displaymath}](img603.png) |
(6) |
Next: Reducing Subspaces.
Up: Singular Case
Previous: Singular Case
  Contents
  Index
Susan Blackford
2000-11-20