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## Conditioning

An eigenvalue of may be well-conditioned or ill-conditioned. If (or is close), the eigenvalues are as well-conditioned (or close) as for the Hermitian eigenproblem described in §2.2.5. But if is very small, where is a unit eigenvector of , then can be very ill conditioned. For example, changing

to

changes an eigenvalue from to ; i.e., the change is magnified by .

Eigenvectors and eigenspaces can also be ill-conditioned for the same reason as in §2.2.5: if the gap between an eigenvalue and the closest other eigenvalue is small, then its eigenvector will be ill-conditioned. Even if the gap is large, if the eigenvalue is ill-conditioned as in the above example, the eigenvector can also be ill-conditioned. Again, an eigenspace spanned by the eigenvectors of a cluster of eigenvalues may be much better conditioned than the individual eigenvectors. We refer to §5.7 for further details.

Next: Specifying an Eigenproblem Up: Generalized Hermitian Eigenproblems   Previous: Eigendecompositions   Contents   Index
Susan Blackford 2000-11-20