Remarks on Clustered Eigenvalues.

As in the case for $B$ being positive definite and well-conditioned, when the eigenvalue $(\alpha,\beta)$ has one or more other eigenvalues of $\{A,B\}$ close by, in other words, when $(\alpha,\beta)$ belongs to a cluster of eigenvalues, as guaranteed by (5.42) the computed $(\wtd\alpha,\wtd\beta)$ is still accurate as long as $\Vert r\Vert _2/\gamma(A,B)$ is tiny, but the computed eigenvector $\wtd x$ may be inaccurate because of the appearance of the gap $\eta$ in the denominator of (5.43). It turns out that each individual eigenvector associated with the clustered eigenvalues is very sensitive to perturbations, but the eigenspace spanned by all the eigenvectors associated with the clustered eigenvalues is not. Thus for the clustered eigenvalues, we should instead compute the entire eigenspace. It can be proved [299,430] that the difference between the computed eigenspace and the eigenspace associated with the cluster is inversely proportional to the gap defined as the smallest difference in chordal metric between any eigenvalue in the cluster and any other eigenvalue not in the cluster. Because of the way it is defined, this gap is expected to be big.