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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