On Solutions of the Matrix Equation A ∘ l X = B with respect to MM -2 Semitensor Product

MM -2 semitensor product is a new and very useful mathematical tool, which breaks the limitation of traditional matrix multiplication on the dimension of matrices and has a wide application prospect. This article aims to investigate the solutions of the matrix equation A ° l X � B with respect to MM -2 semitensor product. The case where the solutions of the equation are vectors is discussed ﬁrst. Compatible conditions of matrices and the necessary and suﬃcient condition for the solvability is studied successively. Furthermore, concrete methods of solving the equation are provided. Then, the case where the solutions of the equation are matrices is studied in a similar way. Finally, several examples are given to illustrate the eﬃciency of the


Introduction
Matrix equations are a very important part of matrix theory [1], and they are often widely applied in many fields. For example, there have been a lot of researches about equations on economic theory [2], automation and information sciences [3][4][5], system and control theory [6][7][8], physics [9,10], and computing sciences [11,12]. For these fields, the numerical approximation solutions [13], least squares solutions [14], symmetric positive [15], and definite solutions [16] under various conditions can be obtained by direct or iterative methods. Furthermore, matrix equations are the basis of numerical calculation and are very good at dealing with arrays that are smaller than three dimensions.
But when dealing with high dimensional data arrangement, power system stability control algebraic, Boolean network, and other problems, the general matrix equation theory is hard to work. Cheng [17] proposed the semitensor product (STP) that successfully improved this shortcoming. As a convenient and powerful mathematical tool, it was quickly studied by scholars and applied to numerical mathematics, control system, Boolean networks, and so on. In 2019, replacing I k in the definition of STP by J k , a new matrix product, called the second matrix-matrix semitensor product (MM-2 STP) of matrices is proposed by Cheng [18]. It provides a new way of solving problems in control systems and has a wide application prospect. For example, cross-dimensional systems are very important dimension-free systems in control theory [19]. ere have been many mathematic models to describe a cross-dimensional system, such as electric power generators [20], spacecrafts [21], and biological systems [22], and switching is a classical method to settle dimension-varying system problems. But it has the disadvantage of neglecting the dynamics of the system in the dimension-varying process [23]. e MM-2 STP could supply a new way to establish unified form models for such switched systems so that we can discuss crossdimensional systems better [24][25][26][27]. Based on this, the solution of the matrix equation A°lX � B with respect to MM-2 semitensor product is studied in this paper.
To achieve the goal, we will study the matrix equation A°lX � B with respect to MM-2 semitensor product by the following steps. At first, the definitions of the second semitensor product of matrices and some other related conceptions are given briefly. After that, the solvability of the matrix equation A°lX � B with respect to MM-2 semitensor product will be discussed in matrix-vector and matrix cases, respectively. In both cases, we give the compatible conditions on matrices A and B first, and then we investigate the necessary and sufficient condition for the solvability. In addition, the concrete steps of solving the equation are clarified. Finally, we give some examples to verify the effectiveness of our results.
ere are 5 sections contained in this article. Section 2 introduces some notations and definitions which will be used later. Section 3 explores the solvability of the matrix-vector equation A°lX � B with respect to MM-2 semitensor product. e compatible conditions of matrices are proposed and the necessary and sufficient condition for the solvability is established. Moreover, concrete solving methods are derived. Section 4 discusses the solvability of the matrix equation A°lX � B with respect to MM-2 semitensor product in the same way. Compatible conditions, solvability conditions, and concrete solving methods of the matrix equation have also been worked out. Section 5 gives some examples and Section 6 draws the conclusion.

Preliminaries
In this article, C n denotes the vector space of complex n-tuples and M m×n denotes the vector space of m × n complex matrices. A T stands for the transpose of a matrix A. m, n { } and gcd m, n { } represent the least common multiple and the greatest common divisor of two positive integers m and n, respectively.
e Kronecker product of matrices A and B, denoted by A ⊗ B, is defined as follows [28]: Definition 2.2. e second left (right) MM-2 semitensor product of matrices A and B, denoted by A°lB(A°rB), is defined as [18] A°lB where t � n, p , J k � (1/k)1 k×k is a k × k matrix with (1/k) as its all entries.

Solution of A ∘ l X = B with X Is a Vector
We now study the solvability of the matrix-vector equation under MM-2 semitensor product, where A � [a ij ] ∈ M m×n , B � [b ij ] ∈ M h×k , and X ∈ C p is an unknown vector to be solved. Initially, we consider the simple case m � h. en, we will discuss the general case.

e Simple Case m � h.
e solvability of the matrixvector equation (4) under the condition that matrices A ∈ M m×n , B ∈ M m×k is studied in this subsection.
At first, similar to the conclusion of Yao in [29], we have the following proposition. Proposition 1. If matrix-vector equation (4) with m � h has a solution, then (n/k) should be a positive integer, and X ∈ C p , p � (n/k).
We call the conditions in Proposition 1 as compatible conditions for matrix-vector equation (4) with m � h. ey are necessary conditions for matrix-vector equation (4). At this time, we say matrices A and B are compatible, and for facility, the matrices A and B are always assumed compatible in the remainder of this subsection.

Theorem 1. Matrix-vector equation (4) with m � h has a solution if and only if €
. . , € A p are linearly independent, the solution would be unique.
Also, the following corollary can be obtained.

Corollary 1. If matrix-vector equation (4) with m � h has a solution, it must satisfy
rough the solving process, we see that equation (5) with m � h is equivalent to the equation as follows: On the other hand, equation (5) can be rewritten as

Journal of Mathematics 3
Let matrix B � (b ij ); it is easy to know that matrix B � (b ij ) must satisfy b ij � b i1 , i � 1, 2, . . . , m, j � 1, 2, . . . , k. at is, B is a matrix in which the elements in the same row are equal. en, we can draw a necessary condition as follows. (4) with m � h has a solution, then matrix B must have the following form:
e matrix B in eorem 2 is said to have the proper form.
Matrix-vector equation (4) with m � h is equivalent to the equation as follows: then we can easily get an equivalent form of matrix-vector equation (4) with m � h. (4) with m � h is equivalent to the matrix-vector equation with conventional matrix product as follows:

Theorem 3. e matrix-vector equation
Also, we can get another corollary.

Corollary 2. If matrix-vector equation (4) with m � h has a solution, the rank condition should satisfy
e solvability condition in eorem 1 is consistent with condition (15). (8) can be rewritten as and (9) can be rewritten as

e General
Case. e solvability of the matrix-vector equation (4) under the condition that matrices A ∈ M m×n , B ∈ M h×k is studied in this subsection.
At first, similar to the conclusion of Yao in [29], we have the following proposition. (4) has a solution, then there will be the following:

Proposition 2. If matrix-vector equation
We call the conditions in Proposition 2 as compatible conditions for matrix-vector equation (4). ey are necessary conditions. At this time, we say matrices A and B are compatible, and for facility, the matrices A and B are always assumed compatible in the remainder of this subsection. Now, we explore the necessary condition for the matrixvector equation first.
□ e matrix B in eorem 4 is said to have the proper form.
en, by the proof of eorem 1, we can draw the following theorem.

Solution of A ∘ l X = B with X Is a Matrix
We now study the solvability of the matrix equation under MM-2 semitensor product, where A � [a ij ] ∈ M m×n , B � [b ij ] ∈ M h×k , and X ∈ M p×q is an unknown matrix to be solved.

e Simple Case m � h.
e solvability of the matrix equation under the condition that matrices A ∈ M m×n , B ∈ M m×k is studied in this subsection.
At first, similar to the conclusion of Yao in [29], we have the following proposition. (24) with m � h has a solution, then X ∈ M p×q , where p � (n/α), q � (k/α), α is a common divisor of n and k.

Proposition 3. If matrix equation
We call the conditions in Proposition 3 as compatible conditions for matrix-vector equation (24) with m � h. ey are necessary conditions for matrix-vector equation (24). At this time, we say matrices A and B are compatible, and for facility, the matrices A and B are always assumed compatible in the remainder of this subsection. Remark 1. Let α i , i � 1, . . . , d to be all the common divisor of n and k. According to Proposition 3, matrix equation (24) with m � h may have solution of size p i × q i , where p i � (n/α i ), q i � (k/α i ), i � 1, . . . , d. At this time, these sizes are called admissible sizes, and we can see some relations between solutions of different admissible sizes.
(1) Let p 1 × q 1 , p 2 × q 2 be two admissible sizes and 1 < (p 2 /p 1 ) � (q 2 /q 1 ) ∈ Z, for the two equations, as follows: If matrix equation (25) has a solution X, then (p 1 /p 2 )(X ⊗ 1 (p 2 /p 1 ) ) is a solution of matrix equation (26); reversely, if matrix equation (26) has unique solution, the solution of equation (25), if exists, would be unique. (2) Let α � gcd n, k { }, p � (n/α), and q � (k/α). If matrix equation (24) with m � h has a minimum size p × q solution, then it has all admissible size solutions. Next, we will study the solvability of matrix equation (24) with m � h. According to Remark 1, the minimum size p × q solutions should be considered first, then the solutions for other admissible sizes can be derived in the same way.

Theorem 7. If matrix-vector equation (24) with m � h has a solution belonging to M p×q , then matrix B can be divided into
q blocks, and furthermore, matrix B must have the following form: ⋮ ⋱ ⋮ ⋮ · · · · · · · · · ⋮ ⋮ ⋱ ⋮ ⋮ · · · · · · · · · ⋮ ⋮ ⋱ ⋮ e matrix B in eorem 7 is said to have the proper form. and vector equation (24) with m � h is equivalent to the equation as follows: x 1i Journal of Mathematics 9 Let W � w 11 w 12 · · · w 1p then we can easily get an equivalent form of matrix-vector equation (24) with m � h.
Theorem 8. e matrix-vector equation (24) with m � h is equivalent to the matrix-vector equation with conventional matrix product as follows: Also, we can get another corollary.

Corollary 4. If matrix-vector equation (24) with m � h has a solution, the rank condition should satisfy
e solvability condition in eorem 7 is consistent with condition (41).

e General
Case. e solvability of the matrix-vector equation (24) under the condition that matrices A ∈ M m×n , B ∈ M h×k is studied in this subsection.
At first, similar to the conclusion of Yao et al. in [29], we have the following proposition. We call the conditions in Proposition 4 as compatible conditions for matrix equation (24). ey are necessary conditions. At this time, we say matrices A and B are compatible, and for facility, the matrices A and B are always assumed compatible in the remainder of this subsection.

Remark 2
(1) e condition m|h in Proposition 4 is just a necessary condition. (2) e sizes in Proposition 4 are called admissible sizes.
When gcd n, k { }|(h/m), it only has one admissible size.
(3) Supposing that p 1 × q 1 , p 2 × q 2 are two admissible sizes and 1 < (p 2 /p 1 ) � (q 2 /q 1 ) ∈ Z, for the two equations, as follows: Similarly, in this case, we can also get a necessary condition for the solvability of matrix equation (24).

Theorem 9. If matrix-vector equation (24) has a solution belonging
to � 1, 2, . . . , m, v � 1, 2, . . . , k) can be divided into m × q blocks, and furthermore, matrix B must have the following form: where Block ij (B) have the form as follows: (46) e matrix B in eorem 9 is said to have the proper form. Now, we give the following algorithm for matrix equation (24).
Step 2. Figure out all the admissible sizes p × q meet the conditions in Proposition 4.
Step 3. For each size p × q, we can solve q matrix-vector equations under MM-2 semitensor product to get the solutions of this matrix equation.

Examples
In this section, we give two cases numerical examples.

Example 1. Considering the matrix-vector equation
A°lX � B, where A and B are as follows (for convenience, we set A ∈ M m×n , B ∈ M h×k , and X ∈ C p .): (1) Noting that m � h, and (n/k) � (4/3), thus the given matrices are not compatible, and by Proposition 1, the equation has no solution.
Noting that m � h, although (n/k) � 2, but B does not have the proper form, so by eorem 2, the equation has no solution. (3) Noting that m � h, (n/k) � 2, and B has the proper form, so by Proposition 1, the equation may have a solution X ∈ C 2 . Let We have taking X � x 1 x 2 T .
(i) Method 1: by definition, we have Solving equation we can get the solution X � 1 2 T , and by eorem 1, the solution is unique. (ii) Method 2: by eorem 3, we have Solving equation we can get the solution X � [12] T , and by eorem 1, the solution is unique.
Noting that m ≠ h, because the given matrices are compatible, B has the proper form, so the equation may have a solution X ∈ C 4 according to Proposition 2. Let X � x 1 x 2 x 3 x 4 T . Comparing with k � l 1 · (h/m) + l 2 , we see l 1 � 1, l 2 � 1, then according to eorem 5, we just need to consider the following equation system: 2x 1 + 2x 3 � 12, Solving it, we get a solution X � 4 2 2 3 T , and X + k 1 − 1 1 1 0 T + k 2 0 − 4 0 1 T , k 1 , k 2 ∈ C} are all the solutions. is also a solution.
Noting that m ≠ h, the given matrices are compatible, and B has the proper form, so the equation may have solutions and the admissible sizes are 3 × 2, 6 × 4 according to Proposition 4. It is easy to verify that In this article, we studied the solutions of the matrix equation A°lX � B with respect to the MM-2 semitensor product. We discussed it in two ways: the solutions are matrices and the solutions are vectors. In each case, we first investigated the necessary conditions for the equation to have a solution. en, we transformed the equation into the equivalent form of ordinary matrix multiplication according to the definition to study the solvability. Further, we obtained the necessary and sufficient conditions for the equation to have a solution and the specific steps to solve the equation. At last, we presented several examples to illustrate the efficiency of our results. We expect the results obtained in this article to be useful. We are sure that they will have broad application prospects in control systems, engineering, computational mathematics, computer science, information science, etc.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.