     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 (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   Index
Susan Blackford 2000-11-20