On the tensor Permutation Matrices

which has the following properties: for any unicolumns and two rows matrices

and for any two 2 × 2-matrices, A, B ∈ C 2×2

This matrix is frequently found in quantum information theory [2], [1], [7].

We call this matrix a tensor commutation matrix (TCM) 2 ⊗ 2. The TCM 3 ⊗ 3 has been written by Kazuyuki Fujii [2] under the following form

in order to obtain a conjecture of the form of a TCM n ⊗ n, for any n ∈ N ⋆ . He calls these matrices “‘swap operator”’. U n⊗p , the TCM n ⊗ p, n, p ∈ N ⋆ , commutes the tensor product of n × n-matrix by p × p-matrix. In this paper we will show that two σ-TPM’s U σ , V σ permute tensor product of rectangle matrices, that is,

, where σ is a permutation of the set {1, 2, . . . , k}. U σ = V σ , if A 1 , A 2 , . . ., A k are square matrices (Cf. for example [5]). We will show this property, according to the Raoelina Andriambololona approach in linear and multilinear algebra [6]: in establishing at first, the propositions on linear operators in intrinsic way, that is independently of the bases, and then we demonstrate the analogous propositions for the matrices.

obtained after the multiplications by scalar, A i j B, is called the tensor product of the matrix A by the matrix B.

A ⊗ B ∈ C mp×nr Proposition 2. Tensor product of matrices is associative.

A is the matrix of A with respect to the couple of bases (e i ) 1 ≤i ≤n , (f j ) 1 ≤j ≤m , B the one of B in (g k ) 1 ≤k ≤r , (h l ) 1 ≤l ≤p . Then, A ⊗ B is the matrix of A ⊗ B with respect to the couple of bases (B, B 1 ), where B = (e 1 ⊗ g 1 , e 1 ⊗ g 2 , . . . , e 1 ⊗ g r , e 2 ⊗ g 1 , e 2 ⊗ g 2 , . . . , e 2 ⊗ g r , . . . , e n ⊗ g 1 , e n ⊗ g 2 , . . . , e n ⊗ g r ) Notation: we denote the set B and B 1 by

), defined by

with respect to the same couple of bases. According to the proposition 5, we have

Proof. " =⇒ " It is evident from the proposition 9. " ⇐= " Suppose that for all

. . , B k be some bases of E 1 , E 2 , . . . , E k where the components of a 1 , a 2 ,. . . , a k form the unicolumn matrices a 1 , a 2 ,. . . ,

The application of this proposition to any two unicolumn matrices leads us to the following remark.

or equivalently

The equation ( 2) can be obtained by multiplying the equation (1) by the m ⊗ q TCM U m⊗q and in using the proposition 9 and the remark 11. Mutually, the equation ( 1) can be obtained by multiplying the equation ( 2) by the q ⊗ m TCM U q⊗m .

We have generalized a property of TPM’s. Two TPM’s permutate tensor product of rectangle matrices. The example show the utility of the property. It suffices to transform a matrix linear equation to a matrix linear equation of the form AX = B. Another matrix linear equation of this form can be deduced by using a TCM and by applying the generalization.