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##

Conditioning

An eigenvalue of a general matrix can be well-conditioned or
ill-conditioned.
For example, if is actually Hermitian (or close to it), its
eigenvalues are well-conditioned as described in §2.2.5.
On the other hand, if is ``far from Hermitian,'' then eigenvalues can be very
ill-conditioned.
For example, matrix in (2.3) shows that
changing a matrix entry by can change the eigenvalues by
, which is much larger than when
.
For example,
is times larger than the
perturbation
, which could be introduced by rounding
error. In other words, the eigenvalues can be perturbed by much
more than the perturbation of
the matrix. As this example hints, this ill-conditioning tends to occur
when two or more eigenvalues are very close together.

The eigenvectors may be similarly well-conditioned or ill-conditioned.
From §2.2.5 we know that close eigenvalues can have
ill-conditioned eigenvectors even for Hermitian matrices. They can even be more
sensitive in the non-Hermitian case, as
in (2.3) again shows:
an perturbation to a matrix whose gap between eigenvalues is
can rotate the eigenvectors by
or even make one of them disappear entirely.

We refer to §7.13 for further details.

** Next:** Specifying an Eigenproblem
** Up:** Non-Hermitian Eigenproblems J. Demmel
** Previous:** Eigendecompositions
** Contents**
** Index**
Susan Blackford
2000-11-20