On the tensor Permutation Matrices

On the Tensor Permutation Matrices

Christian Rakotonirina

September 11, 2018

Département du Génie Civil
Institut Supérieur de Technologie d’Antananarivo, IST-T, BP 8122
Département de Physique
Laboratoire de Rhéologie des Suspensions, LRS, Université d’Antananarivo Madagascar.
e-mail : rakotopierre@refer.mg

Abstract

We show that two TPM’s permute tensor product of rectangle matrices. An example, in the particular case of tensor commutation matrices (TCM’s), for studying a linear matrix equation is given.

Keywords: Tensor product, Matrices, Swap operator, Matrix linear equations.
MCS 2010: 15A69

Introduction

When we were working on Raoelina Andriambololona idea on the using tensor product of matrices in the Dirac equation [4], [8], we met the unitary matrix

$$\mathbf{U}_{2 \otimes 2}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

which has the following properties: for any unicolumns and two rows matrices
$\mathbf{a}=\left[\begin{array}{l}\alpha^{1} \\ \alpha^{2}\end{array}\right] \in \mathbb{C}^{2 \times 1}, \mathbf{b}=\left[\begin{array}{l}\beta^{1} \\ \beta^{2}\end{array}\right] \in \mathbb{C}^{2 \times 1}$

$$\mathbf{U}_{2 \otimes 2} \cdot(\mathbf{a} \otimes \mathbf{b})=\mathbf{b} \otimes \mathbf{a}$$

and for any two $2 \times 2$-matrices, $\mathbf{A}, \mathbf{B} \in \mathbb{C}^{2 \times 2}$

$$\mathbf{U}_{2 \otimes 2} \cdot(\mathbf{A} \otimes \mathbf{B})=(\mathbf{B} \otimes \mathbf{A}) \cdot \mathbf{U}_{2 \otimes 2}$$

This matrix is frequently found in quantum information theory [2], [1], [7]. We call this matrix a tensor commutation matrix (TCM) $2 \otimes 2$. The TCM $3 \otimes 3$ has been written by Kazuyuki Fujii [2] under the following form
in order to obtain a conjecture of the form of a TCM $n \otimes n$, for any $n \in \mathbb{N}^{*}$. He calls these matrices “swap operator”'.
$\mathbf{U}_{n \otimes p}$, the TCM $n \otimes p, n, p \in \mathbb{N}^{*}$, commutes the tensor product of $n \times n$ - matrix by $p \times p$-matrix. In this paper we will show that two $\sigma$ TPM’s $\mathbf{U}_{\sigma}, \mathbf{V}_{\sigma}$ permute tensor product of rectangle matrices, that is, $\mathbf{U}_{\sigma}$. $\left(\mathbf{A}_{1} \otimes \mathbf{A}_{2} \otimes \ldots \otimes \mathbf{A}_{k}\right) \cdot \mathbf{V}_{\sigma}^{T}=\left[\mathbf{A}_{\sigma(1)} \otimes \mathbf{A}_{\sigma(2)} \otimes \ldots \otimes \mathbf{A}_{\sigma(k)}\right.$, where $\sigma$ is a permutation of the set $\{1,2, \ldots, k\}$.
$\mathbf{U}_{\sigma}=\mathbf{V}_{\sigma}$, if $\mathbf{A}_{1}, \mathbf{A}_{2}, \ldots, \mathbf{A}_{k}$ 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.

Tensor Product of Matrices

Definition 1. Consider $\mathbf{A}=\left(A_{j}^{i}\right) \in \mathbb{C}^{m \times n}, \mathbf{B}=\left(B_{j}^{i}\right) \in \mathbb{C}^{p \times r}$. The matrix defined by

$$\mathbf{A} \otimes \mathbf{B}=\left[\begin{array}{cccccc} A_{1}^{1} \mathbf{B} & \ldots & A_{j}^{1} \mathbf{B} & \ldots & A_{n}^{1} \mathbf{B} \\ \vdots & & \vdots & & \vdots \\ A_{1}^{i} \mathbf{B} & \ldots & A_{j}^{i} \mathbf{B} & \ldots & A_{n}^{i} \mathbf{B} \\ \vdots & & \vdots & & \vdots \\ A_{1}^{m} \mathbf{B} & \ldots & A_{j}^{m} \mathbf{B} & \ldots & A_{n}^{m} \mathbf{B} \end{array}\right]$$

obtained after the multiplications by scalar, $A_{j}^{i} \mathbf{B}$, is called the tensor product of the matrix $\mathbf{A}$ by the matrix $\mathbf{B}$.

$$\mathbf{A} \otimes \mathbf{B} \in \mathbb{C}^{m p \times n r}$$

Proposition 2. Tensor product of matrices is associative.
Proposition 3. Consider the linear operators $A \in \mathcal{L}(\mathcal{E}, \mathcal{F}), B \in \mathcal{L}(\mathcal{G}, \mathcal{H})$. $\boldsymbol{A}$ is the matrix of $A$ with respect to the couple of bases $\left(\left(\overline{e_{i}}\right)_{1 \leq i \leq n},\left(\overline{f_{j}}\right)_{1 \leq j \leq m}\right)$, $\boldsymbol{B}$ the one of $B$ in $\left(\left(\overline{g_{k}}\right)_{1 \leq k \leq r},\left(\overline{h_{l}}\right)_{1 \leq l \leq p}\right)$. Then, $\boldsymbol{A} \otimes \boldsymbol{B}$ is the matrix of $A \otimes B$ with respect to the couple of bases $\left(\mathcal{B}, \mathcal{B}_{1}\right)$, where
ParseError: KaTeX parse error: Expected '\right', got 'EOF' at end of input: …{e_{n}} \otimes $\left.\overline{g_{2}}, \ldots, \overline{e_{n}} \otimes \overline{g_{r}}\right)$

Notation: we denote the set $\mathcal{B}$ and $\mathcal{B}_{1}$ by

$$\begin{gathered} \mathcal{B}=\left(\overline{e_{i}} \otimes \overline{g_{k}}\right)_{1 \leq i \leq n, 1 \leq k \leq r}=\left(\left(\overline{e_{i}}\right)_{1 \leq i \leq n}\right) \otimes\left(\left(\overline{g_{k}}\right)_{1 \leq k \leq r}\right) \\ \mathcal{B}_{1}=\left(\overline{f_{j}} \otimes \overline{h_{l}}\right)_{1 \leq j \leq m, 1 \leq l \leq p}=\left(\left(\overline{f_{j}}\right)_{1 \leq j \leq m}\right) \otimes\left(\left(\overline{h_{l}}\right)_{1 \leq l \leq p}\right) \end{gathered}$$

Tensor permutation operators

Definition 4. Consider the $\mathbb{C}$ - vector spaces $\mathcal{E}_{1}, \mathcal{E}_{2}, \ldots, \mathcal{E}_{k}$ and a permutation $\sigma$ of $\{1,2, \ldots, k\}$. The linear operator $U_{\sigma}$ from $\mathcal{E}_{1} \otimes \mathcal{E}_{2} \otimes \ldots \otimes \mathcal{E}_{k}$ to $\mathcal{E}_{\sigma(1)} \otimes \mathcal{E}_{\sigma(2)} \otimes \ldots \otimes \mathcal{E}_{\sigma(k)}, U_{\sigma} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \mathcal{E}_{2} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{E}_{\sigma(1)} \otimes \mathcal{E}_{\sigma(2)} \otimes \ldots \otimes \mathcal{E}_{\sigma(k)}\right.$ ), defined by

$$U_{\sigma}\left(\overline{x_{1}} \otimes \ldots \otimes \overline{x_{k}}\right)=\overline{x_{\sigma(1)}} \otimes \ldots \otimes \overline{x_{\sigma(k)}}$$

for all $\overline{x_{1}} \in \mathcal{E}_{1}, \overline{x_{2}} \in \mathcal{E}_{2}, \ldots, \overline{x_{k}} \in \mathcal{E}_{k}$ is called a $\sigma$-tensor permutation operator (TPO).
If $n=2$, then say that $U_{\sigma}$ is a tensor commutation operator.
Proposition 5. Consider the $\mathbb{C}$ - vector spaces $\mathcal{E}_{1}, \mathcal{E}_{2}, \ldots, \mathcal{E}_{k}, \mathcal{F}_{1}, \mathcal{F}_{2}, \ldots$, $\mathcal{F}_{k}$, a permutation $\sigma$ of $\{1,2, \ldots, k\}$ and a $\sigma$-TPO
$U_{\sigma} \in \mathcal{L}\left(\mathcal{F}_{1} \otimes \ldots \otimes \mathcal{F}_{k}, \mathcal{F}_{\sigma(1)} \otimes \ldots \otimes \mathcal{F}_{\sigma(k)}\right)$. Then, for all $\phi_{1} \in \mathcal{L}\left(\mathcal{E}_{1}, \mathcal{F}_{1}\right)$, $\phi_{2} \in \mathcal{L}\left(\mathcal{E}_{2}, \mathcal{F}_{2}\right), \ldots, \phi_{k} \in \mathcal{L}\left(\mathcal{E}_{k}, \mathcal{F}_{k}\right)$

$$U_{\sigma} \cdot\left(\phi_{1} \otimes \ldots \otimes \phi_{k}\right)=\left(\phi_{\sigma(1)} \otimes \ldots \otimes \phi_{\sigma(k)}\right) \cdot V_{\sigma}$$

where $V_{\sigma} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \mathcal{E}_{2} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{E}_{\sigma(1)} \otimes \mathcal{E}_{\sigma(2)} \otimes \ldots \otimes \mathcal{E}_{\sigma(k)}\right)$ is a $\sigma$-TPO.

Proof. $\phi_{1} \otimes \ldots \otimes \phi_{k} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{F}_{1} \otimes \ldots \otimes \mathcal{F}_{k},\right)$, thus
$U_{\sigma} \cdot\left(\phi_{1} \otimes \phi_{2} \otimes \ldots \otimes \phi_{k}\right) \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{F}_{\sigma(1)} \otimes \ldots \otimes \mathcal{F}_{\sigma(k)}\right)$.
$\left(\phi_{\sigma(1)} \otimes \phi_{\sigma(2)} \otimes \ldots \otimes \phi_{\sigma(k)}\right) \cdot V_{\sigma} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{F}_{\sigma(1)} \otimes \ldots \otimes \mathcal{F}_{\sigma(k)}\right)$
If $\overline{x_{1}} \in \mathcal{E}_{1}, \overline{x_{2}} \in \mathcal{E}_{2}, \ldots, \overline{x_{k}} \in \mathcal{E}_{k}$,
$U_{\sigma} \cdot\left(\phi_{1} \otimes \ldots \otimes \phi_{k}\right)\left(\overline{x_{1}} \otimes \ldots \otimes \overline{x_{k}}\right)$
$=U_{\sigma}\left[\phi_{1}\left(\overline{x_{1}}\right) \otimes \phi_{2}\left(\overline{x_{2}}\right) \otimes \ldots \otimes \phi_{k}\left(\overline{x_{k}}\right)\right]$
$=\phi_{\sigma(1)}\left(\overline{x_{\sigma(1)}}\right) \otimes \ldots \otimes \phi_{\sigma(k)}\left(\overline{x_{\sigma(k)}}\right)$
( Since $U_{\sigma}$ is a TPO)
$=\left(\phi_{\sigma(1)} \otimes \ldots \otimes \phi_{\sigma(k)}\right)\left(\overline{x_{\sigma(1)}} \otimes \ldots \otimes \overline{x_{\sigma(k)}}\right)$
$=\left(\phi_{\sigma(1)} \otimes \ldots \otimes \phi_{\sigma(k)}\right) \cdot V_{\sigma}\left(\overline{x_{1}} \otimes \ldots \otimes \overline{x_{k}}\right)$.
Proposition 6. If $U_{\sigma}$ is a $\sigma$-TPO, then its transpose $U_{\sigma}{ }^{t}$ is the $\sigma^{-1}$-TPO $U_{\sigma^{-1}}$. (Cf. for example [5])

Tensor permutation matrices

Definition 7. Consider the $\mathbb{C}$-vector spaces $\mathcal{E}_{1}, \mathcal{E}_{2}, \ldots, \mathcal{E}_{k}$ of dimensions $n_{1}$, $n_{2}, \ldots, n_{k}$ and $\sigma$-TPO $U_{\sigma} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \mathcal{E}_{2} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{E}_{\sigma(1)} \otimes \mathcal{E}_{\sigma(2)} \otimes \ldots \otimes\right.$ $\left.\mathcal{E}_{\sigma(k)}\right)$. Let
$\mathcal{B}_{1}=\left(\overline{e_{11}}, \overline{e_{12}}, \ldots, \overline{e_{1 n_{1}}}\right)$ be a basis of $\mathcal{E}_{1}$;
$\mathcal{B}_{2}=\left(\overline{e_{21}}, \overline{e_{22}}, \ldots, \overline{e_{2 n_{2}}}\right)$ be a basis of $\mathcal{E}_{2}$;
$\mathcal{B}_{k}=\left(\overline{e_{k 1}}, \overline{e_{k 2}}, \ldots, \overline{e_{k n_{k}}}\right)$ be a basis of $\mathcal{E}_{k}$.
$\mathbf{U}_{\sigma}$ the matrix of $U_{\sigma}$ with respect to the couple of bases
$\left(\mathcal{B}_{1} \otimes \mathcal{B}_{2} \otimes \ldots \otimes \mathcal{B}_{k}, \mathcal{B}_{\sigma(1)} \otimes \mathcal{B}_{\sigma(2)} \otimes \ldots \otimes \mathcal{B}_{\sigma(k)}\right)$. The square matrix $\mathbf{U}_{\sigma}$ of dimensions $n_{1} \times n_{2} \times \ldots \times n_{k}$ is independent of the bases $\mathcal{B}_{1}, \mathcal{B}_{2}, \ldots, \mathcal{B}_{k}$. Call this matrix a $\sigma$-TPM $n_{1} \otimes n_{2} \otimes \ldots \otimes n_{k}$.

According to the proposition 6 , we have the following proposition.
Proposition 8. A $\sigma$-TPM $\boldsymbol{U}_{\sigma}$ is an orthogonal matrix, that is $\boldsymbol{U}_{\sigma}^{-1}=\boldsymbol{U}_{\sigma}^{T}$.
Proposition 9. Let $\boldsymbol{U}_{\sigma}$ be $\sigma$-TPM $n_{1} \otimes n_{2} \otimes \ldots \otimes n_{k}$ and $\boldsymbol{V}_{\sigma}$ a $\sigma$-TPM $m_{1} \otimes m_{2} \otimes \ldots \otimes m_{k}$. Then, for all matrices $\boldsymbol{A}_{1}, \boldsymbol{A}_{2}, \ldots, \boldsymbol{A}_{k}$, of dimensions, respectively, $m_{1} \times n_{1}, m_{2} \times n_{2}, \ldots, m_{k} \times n_{k}$
$\boldsymbol{U}_{\sigma} \cdot\left(\boldsymbol{A}_{1} \otimes \ldots \otimes \boldsymbol{A}_{k}\right) \cdot \boldsymbol{V}_{\sigma}^{T}=\boldsymbol{A}_{\sigma(1)} \otimes \ldots \otimes \boldsymbol{A}_{\sigma(k)}$
Proof. Let $A_{1} \in \mathcal{L}\left(\mathcal{E}_{1}, \mathcal{F}_{1}\right), A_{2} \in \mathcal{L}\left(\mathcal{E}_{2}, \mathcal{F}_{2}\right), \ldots, A_{k} \in \mathcal{L}\left(\mathcal{E}_{k}, \mathcal{F}_{k}\right)$. Their matrices with respect to couple of bases $\left(\mathcal{B}_{1}, \mathcal{B}_{1}^{\prime}\right),\left(\mathcal{B}_{2}, \mathcal{B}_{2}^{\prime}\right), \ldots,\left(\mathcal{B}_{k}, \mathcal{B}_{k}^{\prime}\right)$ are respectively $\mathbf{A}_{1}, \mathbf{A}_{2}, \ldots, \mathbf{A}_{k}$. Then, $\mathbf{A}_{1} \otimes \mathbf{A}_{2} \otimes \ldots \otimes \mathbf{A}_{k}$ is the matrix of

$A_{1} \otimes A_{2} \otimes \ldots \otimes A_{k}$ with respect to $\left(\mathcal{B}_{1} \otimes \mathcal{B}_{2} \otimes \ldots \otimes \mathcal{B}_{k}, \mathcal{B}_{1}^{\prime} \otimes \mathcal{B}_{2}^{\prime} \otimes \ldots \otimes \mathcal{B}_{k}^{\prime}\right)$. But, $A_{1} \otimes A_{2} \otimes \ldots \otimes A_{k} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \mathcal{E}_{2} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{F}_{1} \otimes \mathcal{F}_{2} \otimes \ldots \otimes \mathcal{F}_{k}\right)$ and $A_{\sigma(1)} \otimes A_{\sigma(2)} \otimes \ldots \otimes A_{\sigma(k)} \in \mathcal{L}\left(\mathcal{E}_{\sigma(1)} \otimes \ldots \otimes \mathcal{E}_{\sigma(k)}, \mathcal{F}_{\sigma(1)} \otimes \ldots \otimes \mathcal{F}_{\sigma(k)}\right)$, thus
$U_{\sigma} \cdot\left(A_{1} \otimes \ldots \otimes A_{k}\right),\left(A_{\sigma(1)} \otimes \ldots \otimes A_{\sigma(k)}\right) \cdot V_{\sigma} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{F}_{\sigma(1)} \otimes \ldots \otimes \mathcal{F}_{\sigma(k)}\right)$.
$\mathbf{A}_{\sigma(1)} \otimes \mathbf{A}_{\sigma(2)} \otimes \ldots \otimes \mathbf{A}_{\sigma(k)}$ is the matrix of $A_{\sigma(1)} \otimes A_{\sigma(2)} \otimes \ldots \otimes A_{\sigma(k)}$ with respect to $\left(\mathcal{B}_{\sigma(1)} \otimes \ldots \otimes \mathcal{B}_{\sigma(k)}, \mathcal{B}_{\sigma(1)}^{\prime} \otimes \ldots \otimes \mathcal{B}_{\sigma(k)}^{\prime}\right)$. Thus $\left(\mathbf{A}_{\sigma(1)} \otimes\right.$ $\ldots \otimes \mathbf{A}_{\sigma(k)}) \cdot \mathbf{V}_{\sigma}$ is the one of $\left(A_{\sigma(1)} \otimes \ldots \otimes A_{\sigma(k)}\right) \cdot V_{\sigma}$ with respect to $\left(\mathcal{B}_{1} \otimes \ldots \otimes \mathcal{B}_{k}, \mathcal{B}_{\sigma(1)}^{\prime} \otimes \ldots \otimes \mathcal{B}_{\sigma(k)}^{\prime}\right)$.
$\mathbf{U}_{\sigma} \cdot\left(\mathbf{A}_{1} \otimes \mathbf{A}_{2} \otimes \ldots \otimes \mathbf{A}_{k}\right)$ is the matrix of $U_{\sigma} \cdot\left(A_{1} \otimes A_{2} \otimes \ldots \otimes A_{k}\right)$ with respect to the same couple of bases.
According to the proposition 5 , we have
$\mathbf{U}_{\sigma} \cdot\left(\mathbf{A}_{1} \otimes \ldots \otimes \mathbf{A}_{k}\right)=\left(\mathbf{A}_{\sigma(1)} \otimes \ldots \otimes \mathbf{A}_{\sigma(k)}\right) \cdot \mathbf{V}_{\sigma}$
Proposition 10. The matrix $\boldsymbol{U}_{\sigma}$ is a $\sigma$-TPM $n_{1} \otimes n_{2} \otimes \ldots \otimes n_{k}$ if, and only if, for all $\boldsymbol{a}_{1} \in \mathbb{C}^{n_{1} \times 1}, \boldsymbol{a}_{2} \in \mathbb{C}^{n_{2} \times 1}, \ldots, \boldsymbol{a}_{k} \in \mathbb{C}^{n_{k} \times 1}$
$\boldsymbol{U}_{\sigma} \cdot\left(\boldsymbol{a}_{1} \otimes \ldots \otimes \boldsymbol{a}_{k}\right)=\boldsymbol{a}_{\sigma(1)} \otimes \ldots \otimes \boldsymbol{a}_{\sigma(k)}$
Proof. $" \Longrightarrow "$ It is evident from the proposition 9 .
$" \Longleftarrow "$ Suppose that for all
$\mathbf{a}_{1} \in \mathbb{C}^{n_{1} \times 1}, \mathbf{a}_{2} \in \mathbb{C}^{n_{2} \times 1}, \ldots, \mathbf{a}_{k} \in \mathbb{C}^{n_{k} \times 1}$
$\mathbf{U}_{\sigma} \cdot\left(\mathbf{a}_{1} \otimes \ldots \otimes \mathbf{a}_{k}\right)=\mathbf{a}_{\sigma(1)} \otimes \ldots \otimes \mathbf{a}_{\sigma(k)}$
Let $\overline{a_{1}} \in \mathcal{E}_{1}, \overline{a_{2}} \in \mathcal{E}_{2}, \ldots, \overline{a_{k}} \in \mathcal{E}_{k}$ and $\mathcal{B}_{1}, \mathcal{B}_{2}, \ldots, \mathcal{B}_{k}$ be some bases of $\mathcal{E}_{1}, \mathcal{E}_{2}, \ldots, \mathcal{E}_{k}$ where the components of $\overline{a_{1}}, \overline{a_{2}}, \ldots, \overline{a_{k}}$ form the unicolumn matrices $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{k}$. The $\sigma$-TPO $U_{\sigma} \in \mathcal{L}\left(\mathcal{E}_{1} \otimes \mathcal{E}_{2} \otimes \ldots \otimes \mathcal{E}_{k}, \mathcal{E}_{\sigma(1)} \otimes\right.$ $\mathcal{E}_{\sigma(2)} \otimes \ldots \otimes \mathcal{E}_{\sigma(k)}$ ) whose matrix with respect to $\left(\mathcal{B}_{1} \otimes \mathcal{B}_{2} \otimes \ldots \otimes \mathcal{B}_{k}, \mathcal{B}_{\sigma(1)} \otimes\right.$ $\mathcal{B}_{\sigma(2)} \otimes \ldots \otimes \mathcal{B}_{\sigma(k)}$ ) is $\mathbf{U}_{\sigma}$. Thus
$U_{\sigma}\left(\overline{a_{1}} \otimes \ldots \otimes \overline{a_{k}}\right)=\overline{a_{\sigma(1)}} \otimes \ldots \otimes \overline{a_{\sigma(k)}}$. This is true for all $\overline{a_{1}} \in \mathcal{E}_{1}, \overline{a_{2}} \in \mathcal{E}_{2}$, $\ldots, \overline{a_{k}} \in \mathcal{E}_{k}$.
Since $U_{\sigma}$ is a $\sigma$-TPO, $\mathbf{U}_{\sigma}$ is a $\sigma$-TPM $n_{1} \otimes n_{2} \otimes \ldots \otimes n_{k}$.
The application of this proposition to any two unicolumn matrices leads us to the following remark.

Remark 11. Consider the function $L$ from the set of all matrices into the set of all unicolumn matrices. For a $n \times p$-matrix $\mathbf{X}=\left[\begin{array}{cccc}x_{11} & x_{12} & \ldots & x_{1 p} \\ x_{21} & x_{22} & \ldots & x_{2 p} \\ \ldots & \ldots & \ldots & \ldots \\ x_{n 1} & x_{n 2} & \ldots & x_{n p}\end{array}\right]$,

$L(\mathbf{X})=\left[\begin{array}{c}x_{11} \\ x_{12} \\ \vdots \\ x_{1 p} \\ x_{21} \\ x_{22} \\ \vdots \\ x_{2 p} \\ \vdots \\ x_{n 1} \\ x_{n 2} \\ \vdots \\ x_{n p}\end{array}\right]$. The relation $\mathbf{U}_{n \otimes p} \cdot L(\mathbf{X})=L\left(\mathbf{X}^{T}\right)$ can be obtained easily.
Example 12. Consider the matrix equation $\mathbf{A} \cdot \mathbf{X} \cdot \mathbf{B}=\mathbf{C}$, with respect to the unknown $\mathbf{X} \in \mathbb{C}^{n \times q}$, where $\mathbf{A} \in \mathbb{C}^{m \times n}, \mathbf{B} \in \mathbb{C}^{p \times q}$ and $\mathbf{C} \in \mathbb{C}^{m \times q}$ are known. This equation can be transformed to the system of linear equations, whose matrix equation is [3]

$$\left(\mathbf{A} \otimes \mathbf{B}^{T}\right) \cdot L(\mathbf{X})=L(\mathbf{C})$$

or equivalently

$$\left(\mathbf{B}^{T} \otimes \mathbf{A}\right) \cdot L\left(\mathbf{X}^{T}\right)=L\left(\mathbf{C}^{T}\right)$$

The equation (2) can be obtained by multiplying the equation (1) by the $m \otimes q \operatorname{TCM} \mathbf{U}_{m \otimes q}$ and in using the proposition 9 and the remark 11.
Mutually, the equation (1) can be obtained by multiplying the equation (2) by the $q \otimes m \operatorname{TCM} \mathbf{U}_{q \otimes m}$.

Conclusion

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 $\mathbf{A X}=\mathbf{B}$. Another matrix linear equation of this form can be deduced by using a TCM and by applying the generalization.

Acknowledgments

The author would like to thank Andriamifidisoa Ramamonjy of the Department of Mathematics and Informatics of the University of Antananarivo for

his help in preparing the manuscript.

References

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