Structure-preserving diagonalization of matrices in indefinite inner product spaces

TL;DR

This paper investigates conditions under which matrices in indefinite inner product spaces can be diagonalized while preserving their structure, providing criteria for symplectic and perplectic diagonalizations, especially for normal matrices, with practical examples.

Contribution

It establishes necessary and sufficient conditions for structure-preserving diagonalization of (skew)-Hamiltonian and (per(skew))-Hermitian matrices, including the case of normal matrices with unitary diagonalizations.

Findings

Conditions for symplectic diagonalizability of Hamiltonian matrices

Conditions for perplectic diagonalizability of Hermitian matrices

Existence of unitary, structure-preserving diagonalizations for normal matrices

Abstract

In this work some results on the structure-preserving diagonalization of selfadjoint and skewadjoint matrices in indefinite inner product spaces are presented. In particular, necessary and sufficient conditions on the symplectic diagonalizability of (skew)-Hamiltonian matrices and the perplectic diagonalizability of per(skew)-Hermitian matrices are provided. Assuming the structured matrix at hand is additionally normal, it is shown that any symplectic or perplectic diagonalization can always be constructed to be unitary. As a consequence of this fact, the existence of a unitary, structure-preserving diagonalization is equivalent to the existence of a specially structured additive decomposition of such matrices. The implications of this decomposition are illustrated by several examples.