On the Existence of Schur-like Forms for Matrices with Symmetry Structures

Abstract

AbstractSchur-like forms are developed for matrices that have a symmetry structure with respect to an indefinite inner product induced by a Hermitian and unitary Gram matrix. It is characterized under which conditions these forms can be computed by structure-preserving unitary transformations. The main results combines and generalizes the two well-known results from the literature that on the one hand any normal matrix can be unitarily diagonalized and on the other hand a Hamiltonian matrix can be transformed to Hamiltonian Schur form via a unitary similarity transformation if and only if its purely imaginary eigenvalues satisfy certain conditions that involve the sign characteristic of the matrix under consideration.