Conditioning

An eigenvalue $\lambda_i$ of $A - \lambda B$ may be well-conditioned or ill-conditioned. If $B=I$ (or is close), the eigenvalues are as well-conditioned (or close) as for the Hermitian eigenproblem described in §2.2.5. But if $\vert x_i^* B x_i\vert$ is very small, where $x_i$ is a unit eigenvector of $\lambda_i$, then $\lambda_i$ can be very ill conditioned. For example, changing

$$A_0 - \lambda B_0 = \bmat{ccc} 2 & 0 \\ 0 & 10^{-6} \\ \emat - \lambda \bmat{ccc} 1 & 0 \\ 0 & 10^{-6} \\ \emat$$

to

$$A_1 - \lambda B_1 = \bmat{ccc} 2 & 0 \\ 0 & 10^{-6} + \epsilon \\ \emat - \lambda \bmat{ccc} 1 & 0 \\ 0 & 10^{-6} \\ \emat$$

changes an eigenvalue from $1 = 10^{-6}/10^{-6}$ to $(10^{-6} + \epsilon)/10^{-6} = 1 + 10^6 \epsilon$; i.e., the $\epsilon$ change is magnified by $10^6$.

Eigenvectors and eigenspaces can also be ill-conditioned for the same reason as in §2.2.5: if the gap between an eigenvalue and the closest other eigenvalue is small, then its eigenvector will be ill-conditioned. Even if the gap is large, if the eigenvalue is ill-conditioned as in the above example, the eigenvector can also be ill-conditioned. Again, an eigenspace spanned by the eigenvectors of a cluster of eigenvalues may be much better conditioned than the individual eigenvectors. We refer to §5.7 for further details.