Isotropy groups of the action of orthogonal similarity on skew-symmetric and on complex orthogonal matrices

TL;DR

This paper investigates the structure of isotropy groups under orthogonal similarity transformations, focusing on complex orthogonal and skew-symmetric matrices, revealing their connection to block Toeplitz matrices.

Contribution

It provides a detailed analysis of isotropy subgroups for complex orthogonal similarity actions, linking their structure to block Toeplitz matrices, which is a novel insight.

Findings

Characterization of isotropy subgroups for complex orthogonal similarity

Connection between isotropy groups and block Toeplitz matrices

Structural analysis of group actions on skew-symmetric matrices

Abstract

We compute and analyze isotropy subgroups of the complex orthogonal group with respect to the similarity transformation on itself and on skew-symmetric matrices. Their group structure is related to a group of certain nonsingular block matrices whose blocks are rectangular block Toeplitz.