Commutation matrices and Commutation tensors

TL;DR

This paper explores the properties of commutation matrices and tensors, extending classical results in matrix rank preservation and unifying different formulae through tensor generalization.

Contribution

It introduces the concept of commutation tensors and applies them to unify two classical formulas related to matrix rank preservers.

Findings

Extended commutation matrix to commutation tensor.

Unified two classical formulas of matrix rank preservers.

Provided new insights into linear transformations involving transposition.

Abstract

The commutation matrix was first introduced in statistics as a transposition matrix by Murnaghan in 1938. In this paper, we first investigate the commutation matrix which is employed to transform a matrix into its transpose. We then extend the concept of the commutation matrix to commutation tensor and use the commutation tensor to achieve the unification of the two formulae of the linear preserver of the matrix rank, a classical result of Marcus in 1971.