Similarity vs unitary similarity and perturbation analysis of sign characteristics: Complex and real indefinite inner products

For selfadjoint and unitary matrices with respect to indefinite inner products on finite dimensional complex or real vector spaces, the unitary similarity relation is studied in contrast to the similarity relation. The number of unitary similarity classes in a similarity class of such matrices is identified. It is proved that under sufficiently small perturbations, similarity of selfadjoint or unitary (with respect to an indefinite inner product) matrices implies unitary similarity. Estimates are given for the sizes of perturbation that guarantee this property. Canonical forms and sign characteristics play a key role in the proofs. A perturbation analysis is developed of the sign characteristics in the real case, under sufficiently small structure preserving perturbations of both selfadjoint or unitary matrices and of the indefinite inner product.