Tensor Permutation Matrices in Finite Dimensions

TL;DR

This paper introduces tensor permutation matrices in finite dimensions, generalizing tensor commutation matrices, with formulas for construction and expressions, including representations using Gell-Mann matrices.

Contribution

It generalizes tensor commutation matrices to tensor permutation matrices and provides formulas and expressions for their construction and elements.

Findings

Established a formula for tensor permutation matrices.

Expressed tensor commutation matrices using Gell-Mann matrices.

Generalized element expressions for tensor permutation matrices.

Abstract

We have generalised the properties with the tensor product, of one 4x4 matrix which is a permutation matrix, and we call a tensor commutation matrix. Tensor commutation matrices can be constructed with or without calculus. A formula allows us to construct a tensor permutation matrix, which is a generalisation of tensor commutation matrix, has been established. The expression of an element of a tensor commutation matrix has been generalised in the case of any element of a tensor permutation matrix. Some tensor commutation matrix has been expressed by using the Gell-Mann matrices.