ON THE COMMUTATION MATRIX

ABSTRACT


INTRODUCTION
The  matrix is a unique operation that can change a matrix into a column vector [1].It can also be said to change the matrix into a vector by stacking the column vertically [2].Note that () and (  ) have duplicate entries, but the composition of the elements is different.If the matrix ,  × , and its transpose is matrix   , then the vectors () and (  ) are  × 1.The  operator then inspires the creation of new operators with similar transformations, making the matrix a column vector.Operators who were later born were ℎ, , , and  operators (see [3], [4], [5], [6], and [7]).With the new operators after the  operator, the relation between each operator is also created.
A unique matrix that transforms () to (  ) for any matrix  ×  [2] is called by commutation matrix.This matrix is defined as a square matrix containing only zeroes and ones.In the previous study, the commutation matrix can be connected to the statistics, i.e., the matrix is applied to some problems related to normal distributions [8].Furthermore, commutation matrix establishes the relation between the Kronecker product and the vec-permutation matrix [9].Then, [10] extends the concept of the commutation matrix to the commutation tensor and uses the commutation tensor to achieve the unification of the two formulas of linear preserver of the matrix rank.
In [11], it is stated that there are matrices that are like a commutation matrix, i.e., the matrix transform the  matrix  to the  transpose matrix  ( ∈ ℂ × ) for the matrices in the Kronecker quaternion group found in [12].Similar properties are duplicate entries on matrix  in the Kronecker quaternion group with the same position.
The previous study obtained the commutation matrix result using properties of vec some matrices (two or more).In this paper, we present the proof of some properties of the commutation matrix differently.
The organization of this paper is as follows.In the Research Methods, some basic concepts and notations of , permutation matrix, Kronecker product, and commutation matrix will be used in the section Result and Discussion, are presented.In the section Result and Discussion, the definitions of commutation matrix are discussed.Next, it presents the properties of the commutation matrix.

RESEARCH METHODS
The research methods are based on the literature study related to the commutation matrix.The first step of this research is to show the equivalence of the three definitions and then some theorems related to the commutation matrix.
First, this section presents some definitions, properties and theorems related to commutation matrix, i.e., vec, permutation matrix, and Kronecker product.

Definition 1 [13]
Let  = [  ] be an  ×  matrix, and   is the column of .The () is the -column vector, i.e., Or it can also be written with () = There is another notation commonly used to specify permutation.It is called cycle notation.Cycle notation has theoretical advantages in that specific essential properties of the permutation can be readily determined when cycle notation is used.For example, permutation in Equation (1) can be written as  = (1 7 5 2)(4 6).For detail, see [14].
If  is a permutation, we have  change the identity matrix as follows: Definition 2. [15] Let  be a permutation in   .Define the permutation matrix () = [ ,() ] ,  ,() =  , (()), where Let  be a arbitrary  ×  matrix, the commutation matrix of  is a matrix that transform  matrix  to the  transpose matrix.There are several ways to define this matrix, and in this paper is given three different ways to determine this commutation matrix.Definition 4.

a. [13]
Let   be an  ×  matrix with 1 in its (, ) th position and zero elsewhere.Then the  ×  commutation matrix, denoted by   , is given by: Remark.The matrix   can be conveniently expressed using the column from the identity matrices   and   .If  , is the i th column of   , and  , is the j th column of   , then   =  ,  ,  .
The commutation matrix, denoted by  , is given by:

c. [10]
A permutation matrix  is called a commutation matrix of a matrix,  × , if it satisfies the following conditions: i.  = [  ] is an  ×  block matrix, with each block   being an  ×  matrix.
We denote this commutation matrix by  , , thus a commutation matrix is of size  × .
Example 3. Using Definition 4 (ac), we demonstrate how to create the commutation matrix.

RESULTS AND DISCUSSION
In this section, we present the properties of the commutation matrix.Since, there are three ways to determine the commutation matrix, so in Theorem 7 is given that the three definitions are equivalent.Furthermore, in Theorem 8, is proven that the commutation matrix is the same as its transpose.And last, in Theorem 9 it is proven that relation between  matrix and  transpose matrix.
(2 ⇒ 3) Consider that We have that  , is a matrix  ×  with i.  = [  ] is a  ×  block matrix with each block   being a  ×  matrix.

CONCLUSIONS
This paper establishes some conclusions on the commutation matrix.The results of the properties of the commutation matrix using one of the definitions, the commutation matrix.A different way to prove the properties of the commutation matrix is given.All these obtained conclusions make the theory of commutation matrix more complete.