The direct product permuting matrices

A new matrix product is defined and its properties are investigated. The commutatuion matrix which flips a left direct product of two matrices into a right direct one is derived as a composition of two identity matrices. The communication matrix is a special case of the direct product permuting matrices defined in this paper which are matrix representations of the permutation operators on tensor spaces i e. the linear mappings which permute the order of the vectors in a direct product of them. Explicit expressions for these matrices are given. properties of the matrices are investigated and it is shown how these matrices, act on various representations of tensor spaces.