The Cyclic Group and the Transpose of an R-cyclic matrix

TL;DR

This paper explores how the cyclic group can be used to decompose the transpose of an R-cyclic matrix into freely independent parts, revealing new structural properties especially for self-adjoint matrices.

Contribution

It introduces a novel decomposition method for the transpose of R-cyclic matrices using cyclic groups, highlighting independence and R-diagonality properties.

Findings

Decomposition of transpose into freely independent diagonal parts

Off-diagonal parts are R-diagonal for self-adjoint matrices

Provides structural insights into R-cyclic matrices

Abstract

We show that using the cyclic group the transpose of an R-cyclic matrix can be decomposed along diagonal parts into a sum of parts which are freely independent over diagonal scalar matrices. Moreover, if the R-cyclic matrix is self-adjoint then the off-diagonal parts are R-diagonal.