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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

(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

(5) 
Then is an eigenvalue but
for any for any value of .
Instead we consider reducing subspaces, as defined below.

(6) 
Next: Reducing Subspaces.
Up: Singular Case
Previous: Singular Case
Contents
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Susan Blackford
20001120