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.