The Rotation of Eigenvectors by a Perturbation. III

When a Hermitian linear operator is slightly perturbed, by how much can its invariant subspaces change? Given some approximations to a cluster of neighboring eigenvalues and to the corresponding eigenvectors of a real symmetric matrix, and given an estimate for the gap that separates the cluster from all other eigenvalues, how much can the subspace spanned by the eigenvectors differ from the subspace spanned by our approximations? These questions are closely related; both are investigated here. The difference between the two subspaces is characterized in terms of certain angles through which one subspace must be rotated in order most directly to reach the other. These angles unify the treatment of natural geometric, operator-theoretic and error-analytic questions concerning those subspaces. Sharp bounds upon trigonometric functions of these angles are obtained from the gap and from bounds upon either the perturbation (1st question) or a computable residual (2nd question). An example is included.