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Wang, Jin

doi:https://doi.org/10.1155/2021/6651434

Journal of Mathematics

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Abstract Introduction Preliminaries Examples Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6651434 | https://doi.org/10.1155/2021/6651434

On Solutions of the Matrix Equation = B with respect to -2 Semitensor Product

Jin Wang¹

Academic Editor: Efthymios G. Tsionas

Received14 Dec 2020

Revised27 Jan 2021

Accepted29 Jan 2021

Published24 Feb 2021

Abstract

-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 with respect to -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 results.

1. 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 [3–5], system and control theory [6–8], 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 in the definition of STP by , a new matrix product, called the second matrix-matrix semitensor product (-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]. There 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]. The -2 STP could supply a new way to establish unified form models for such switched systems so that we can discuss cross-dimensional systems better [24–27]. Based on this, the solution of the matrix equation with respect to -2 semitensor product is studied in this paper.

To achieve the goal, we will study the matrix equation with respect to -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 with respect to -2 semitensor product will be discussed in matrix-vector and matrix cases, respectively. In both cases, we give the compatible conditions on matrices and 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.

There 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 with respect to -2 semitensor product. The 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 with respect to -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.

2. Preliminaries

In this article, denotes the vector space of complex -tuples and denotes the vector space of complex matrices. stands for the transpose of a matrix . and represent the least common multiple and the greatest common divisor of two positive integers and , respectively.

Definition 2.1. The Kronecker product of matrices and , denoted by , is defined as follows [28]:

Definition 2.2. The second left (right) -2 semitensor product of matrices and , denoted by , is defined as [18]where is a matrix with as its all entries.

Definition 2.3. The vectorization of , denoted by , is defined as [28]

3. Solution of = B with Is a Vector

We now study the solvability of the matrix-vector equation under -2 semitensor product,where , and is an unknown vector to be solved.

Initially, we consider the simple case . Then, we will discuss the general case.

3.1. The Simple Case

The solvability of the matrix-vector equation (4) under the condition that matrices 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 has a solution, then should be a positive integer, and .

We call the conditions in Proposition 1 as compatible conditions for matrix-vector equation (4) with . They are necessary conditions for matrix-vector equation (4). At this time, we say matrices and are compatible, and for facility, the matrices and are always assumed compatible in the remainder of this subsection.

By Proposition 1, if matrix-vector equation (4) with has a solution, then . Therefore, we should inspect whether there is a -dimension vector that satisfies matrix-vector (4) with . Let . Then, according to the definition, we havewhere are the equal-size blocks of matrix , and

Let be the sum of all the elements in row of , then

On the one hand, the following theorem can be obtained.

Theorem 1. Matrix-vector equation (4) with has a solution if and only if and are linearly dependent in vector space . Furthermore, when are linearly independent, the solution would be unique.

Also, the following corollary can be obtained.

Corollary 1. If matrix-vector equation (4) with has a solution, it must satisfy

Through the solving process, we see that equation (5) with is equivalent to the equation as follows:

On the other hand, equation (5) can be rewritten as

Let matrix ; it is easy to know that matrix must satisfy . That is, is a matrix in which the elements in the same row are equal.

Then, we can draw a necessary condition as follows.

Theorem 2. If matrix-vector equation (4) with has a solution, then matrix must have the following form:where denotes the -th column of .

The matrix in Theorem 2 is said to have the proper form.

Matrix-vector equation (4) with m = h is equivalent to the equation as follows:

Letthen we can easily get an equivalent form of matrix-vector equation (4) with .

Theorem 3. The matrix-vector equation (4) with is equivalent to the matrix-vector equation with conventional matrix product as follows:

Also, we can get another corollary.

Corollary 2. If matrix-vector equation (4) with has a solution, the rank condition should satisfy

The solvability condition in Theorem 1 is consistent with condition (15).

At the same time, let to be the -th column of , then (8) can be rewritten asand (9) can be rewritten as

3.2. The General Case

The solvability of the matrix-vector equation (4) under the condition that matrices is studied in this subsection.

At first, similar to the conclusion of Yao in [29], we have the following proposition.

Proposition 2. If matrix-vector equation (4) has a solution, then there will be the following: (i) and are positive integers; (ii) , and the solution .

We call the conditions in Proposition 2 as compatible conditions for matrix-vector equation (4). They are necessary conditions. At this time, we say matrices and are compatible, and for facility, the matrices and are always assumed compatible in the remainder of this subsection.

Now, we explore the necessary condition for the matrix-vector equation first.

Theorem 4. If matrix-vector equation (4) has a solution, then matrix must have the following form:whereThat is,

Proof. According to Proposition 2, the solution of matrix-vector equation (4) belongs to , as ; let , where are integers. Thus, for , we can getThus, are matrices having the form as follows:The proof is completed.

The matrix in Theorem 4 is said to have the proper form.

Then, by the proof of Theorem 1, we can draw the following theorem.

Theorem 5. Solving matrix-vector equation (4) is equivalent to solving the following equation system:

4. Solution of = B with Is a Matrix

We now study the solvability of the matrix equation under -2 semitensor product,where , and is an unknown matrix to be solved.

4.1. The Simple Case

The solvability of the matrix equation under the condition that matrices is studied in this subsection.

At first, similar to the conclusion of Yao in [29], we have the following proposition.

Proposition 3. If matrix equation (24) with has a solution, then , where is a common divisor of and .

We call the conditions in Proposition 3 as compatible conditions for matrix-vector equation (24) with . They are necessary conditions for matrix-vector equation (24). At this time, we say matrices and are compatible, and for facility, the matrices and are always assumed compatible in the remainder of this subsection.

Remark 1. Let to be all the common divisor of and . According to Proposition 3, matrix equation (24) with may have solution of size , where . At this time, these sizes are called admissible sizes, and we can see some relations between solutions of different admissible sizes.(1)Let be two admissible sizes and , for the two equations, as follows: If matrix equation (25) has a solution , then 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 , and . If matrix equation (24) with has a minimum size solution, then it has all admissible size solutions.(3)If , matrix equation (24) with will have only one admissible size , and at this time, it is a conventional one.(4)If , every admissible size solution, if exists, would be unique.

Next, we will study the solvability of matrix equation (24) with . According to Remark 1, the minimum size solutions should be considered first, then the solutions for other admissible sizes can be derived in the same way.

Let . By the definition, matrix equation (24) with can be rewritten aswhere are equal-size blocks of matrix , . Thus, matrix equation (24) with is equivalent to the following matrix-vector equations under -2 semitensor product:

Aswhere are the equal-size blocks of matrix , belonging to , and

Let be the sum of all the elements in row of , then

Thus, on the one hand, we have the following necessary and sufficient condition.

Theorem 6. For matrix equation (24) with , it has a solution belonging to if and only if and are linearly dependent in vector space . Furthermore, when are linearly independent, the solution would be unique.

Simultaneously, we can get a necessary but not sufficient condition.

Corollary 3. If matrix-vector equation (24) with has a solution belonging to , it must satisfy

On the other hand, equation (29) can be rewritten as.

By equation (33), it is easy to know that are matrices in which the elements in the same row are equal. Let denote the -th column of . That is,

Letand matrix . By comparison, we can get , and therefore, we have

Then, we can draw a necessary condition as follows.

Theorem 7. If matrix-vector equation (24) with has a solution belonging to , then matrix can be divided into blocks, and furthermore, matrix must have the following form:

The matrix in Theorem 7 is said to have the proper form. and vector equation (24) with is equivalent to the equation as follows:

Letthen we can easily get an equivalent form of matrix-vector equation (24) with .

Theorem 8. The 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 has a solution, the rank condition should satisfy

The solvability condition in Theorem 7 is consistent with condition (41).

At the same time, let to be the -th column of , then equations (32) and (40) can be rewritten as

4.2. The General Case

The solvability of the matrix-vector equation (24) under the condition that matrices is studied in this subsection.

At first, similar to the conclusion of Yao et al. in [29], we have the following proposition.

Proposition 4. If matrix equation (24) has a solution, then there will be the following: (i) is a positive integer; (ii) the solution , where is a common divisor of and , which satisfies .

We call the conditions in Proposition 4 as compatible conditions for matrix equation (24). They are necessary conditions. At this time, we say matrices and are compatible, and for facility, the matrices and are always assumed compatible in the remainder of this subsection.

Remark 2. (1)The condition in Proposition 4 is just a necessary condition.(2)The sizes in Proposition 4 are called admissible sizes. When , it only has one admissible size. At this time, we see that , and matrix equation (14) is just a conventional matrix equation .(3)Supposing that are two admissible sizes and , for the two equations, as follows:(4) If matrix equation (43) has a solution , then is a solution of matrix equation (44); reversely, if equation (44) has unique solution, the solution of equation (43), if exists, would be unique.(5)Denote . If matrix equation (24) has a minimum size solution, then it has all admissible size solutions.(6)If , every admissible size solution, if exists, would be unique.

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 , then matrix can be divided into blocks, and furthermore, matrix must have the following form:where have the form as follows:

The matrix in Theorem 9 is said to have the proper form.

Now, we give the following algorithm for matrix equation (24).

Step 1. To see whether matrix equation (24) suits the compatible conditions or not, that is, and , in the proper form,where have the form as .

Step 2. Figure out all the admissible sizes meet the conditions in Proposition 4.

Step 3. For each size , we can solve matrix-vector equations under -2 semitensor product to get the solutions of this matrix equation.

5. Examples

In this section, we give two cases numerical examples.

Example 1. Considering the matrix-vector equation , where and are as follows (for convenience, we set , and .):(1) Noting that , and , thus the given matrices are not compatible, and by Proposition 1, the equation has no solution.(2) Noting that , although , but does not have the proper form, so by Theorem 2, the equation has no solution.(3) Noting that , and has the proper form, so by Proposition 1, the equation may have a solution . Let We have taking .(i)Method 1: by definition, we have Solving equation we can get the solution , and by Theorem 1, the solution is unique.(ii) Method 2: by Theorem 3, we have Solving equation we can get the solution , and by Theorem 1, the solution is unique.(4) Noting that , the given matrices are compatible and has the proper form; therefore, by Proposition 1, we know the equation may have a solution . But through calculating, we can see , so this equation has no solution according to Corollary 2.(5) Noting that , are positive integers, but does not have the proper form. Hence, by Theorem 4, the equation has no solution.(6) Noting that , because the given matrices are compatible, has the proper form, so the equation may have a solution according to Proposition 2. Let . Comparing with , we see , then according to Theorem 5, we just need to consider the following equation system: Solving it, we get a solution , and are all the solutions.

Example 2. Considering the matrix equation , where and are as follows (for convenience, we set , and .):(1) Noting that , as does not have the proper form, hence the equation has no solution according to Theorem 7.(2) Noting that , and has the proper form, hence the equation may have solutions and the admissible sizes are according to Proposition 3. It is easy to verify that is a solution. Thus, by Remark 1, is also a solution.(3) Noting that , the given matrices are compatible, and has the proper form, so the equation may have solutions and the admissible sizes are according to Proposition 4. It is easy to verify that is a solution. Thus, by Remark 2, is also a solution.

6. Conclusion

In this article, we studied the solutions of the matrix equation with respect to the -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. Then, 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

The author declares that there are no conflicts of interest.

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Copyright © 2021 Jin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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