Conditioning

An eigenvalue of a general matrix $A$ can be well-conditioned or ill-conditioned. For example, if $A$ is actually Hermitian (or close to it), its eigenvalues are well-conditioned as described in §2.2.5. On the other hand, if $A$ is ``far from Hermitian,'' then eigenvalues can be very ill-conditioned. For example, matrix $A(\epsilon)$ in (2.3) shows that changing a matrix entry by $\epsilon^2$ can change the eigenvalues by $\epsilon$, which is much larger than $\epsilon^2$ when $\vert\epsilon\vert \ll 1$. For example, $\epsilon = 10^{-8}$ is $10^8$ times larger than the perturbation $\epsilon^2 = 10^{-16}$, 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 $A(\epsilon)$ in (2.3) again shows: an $\epsilon^2$ perturbation to a matrix whose gap between eigenvalues is $2 \epsilon$ can rotate the eigenvectors by $\epsilon \gg \epsilon^2$ or even make one of them disappear entirely.

We refer to §7.13 for further details.