Isotropy groups of the action of orthogonal similarity on symmetric matrices

TL;DR

This paper presents an algorithmic method to compute and describe the isotropy subgroups of complex orthogonal matrices acting on symmetric matrices, involving solving a specific Toeplitz matrix equation.

Contribution

It introduces a novel algorithmic approach for characterizing isotropy subgroups under orthogonal similarity actions on symmetric matrices.

Findings

Developed an algorithm to compute isotropy subgroups.

Provided a structural description of these subgroups.

Solved a key Toeplitz matrix equation integral to the analysis.

Abstract

We find an algorithmic procedure that enables to compute and to describe the structure of the isotropy subgroups of the group of complex orthogonal matrices with respect to the action of similarity on complex symmetric matrices. A key step in our proof is to solve a certain rectangular block upper-triangular Toeplitz matrix equation.