Isotropy groups of the action of orthogonal *congruence on Hermitian matrices

TL;DR

This paper develops a method to compute isotropy groups of complex orthogonal matrices acting on Hermitian matrices via *congruence, utilizing an algorithm for solving specific block Toeplitz matrix equations.

Contribution

It introduces a procedure for describing isotropy subgroups under *congruence action, including an algorithm for solving related block Toeplitz matrix equations.

Findings

Provides a systematic way to compute isotropy groups.

Describes the structure of isotropy subgroups explicitly.

Introduces an algorithm for solving complex block Toeplitz equations.

Abstract

We present a procedure which enables the computation and the description of structures of isotropy subgroups of the group of complex orthogonal matrices with respect to the action of *congruence on Hermitian matrices. A key ingredient in our proof is an algorithm giving solutions of a certain rectangular block (complex-alternating) upper triangular Toeplitz matrix equation.