Conditioning

The eigenvalues of $A$ are always well-conditioned, in the sense that changing $A$ in norm by at most $\epsilon$ can change any eigenvalue by at most $\epsilon$. We refer to §4.8 for technical definitions.

This is adequate for most purposes, unless the user is interested in the leading digits of a small eigenvalue, one less than or equal to $\epsilon$ in magnitude. For example, computing $\lambda_i = 10^{-5}$ to within plus or minus $\epsilon = 10^{-4}$ means that no leading digits of the computed $\lambda_i$ may be correct. See [114,118] for a discussion of the sensitivity of small eigenvalues and of when their leading digits may be computed accurately.

Eigenvectors and eigenspaces, on the other hand, can be ill-conditioned. For example, changing

$$A_0 = \bmat{ccc} 2 & 0 & 0 \\ 0 & 1 & \epsilon \\ 0 & \epsil... ... \bmat{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1+\epsilon \emat$$

rotates the two eigenvectors corresponding to the two eigenvalues near 1 by $\pi/4$, no matter how small $\epsilon$ is. Thus they are very sensitive to small changes. The condition number of an eigenvector depends on the gap between its eigenvalue and the closest other eigenvalue: the smaller the gap the more sensitive the eigenvectors. In this example the two eigenvalues near 1 are very close, so their gaps are small and their eigenvectors are sensitive. But the two-dimensional invariant subspace they span is very insensitive to changes in $A$ (because their eigenvalues, both near 1, are very far from the next closest eigenvalue, at 2). So when eigenvectors corresponding to a cluster of close eigenvalues are too ill-conditioned, the user may want to compute a basis of the invariant subspace they span instead of individual eigenvectors.