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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