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For nonnormal matrices the norms of the residuals of approximate eigenvectors are not by themselves sufficient information to bound the error in the approximate eigenvalue. It is sufficient however to give a bound on the distance to the nearest matrix for which the given approximations are exact. This result is extended to cover approximate invariant subspaces and their residuals. The theorems are used to derive a useful set of criteria for terminating the two-sided Lanczos algorithm. The study begins with a list of error bounds for eigenvalues of nonnormal matrices.
Kahan et al. (1982) studied this question.