Some Further Results on the Reduction of Two-Dimensional Systems

Li, Dongmei; Gui, Yingying; Liu, Jinwang; Wu, Man

doi:https://doi.org/10.1155/2021/6911443

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Behaviour Analysis, Complexity and Control of Networked Dynamical Systems

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Volume 2021 | Article ID 6911443 | https://doi.org/10.1155/2021/6911443

Some Further Results on the Reduction of Two-Dimensional Systems

Dongmei Li,¹Yingying Gui,¹Jinwang Liu,¹and Man Wu¹

Academic Editor: Xiao Ling Wang

Received05 Aug 2021

Accepted29 Oct 2021

Published17 Nov 2021

Abstract

The reduction of two-dimensional systems plays an important role in the theory of systems, which is closely associated with the equivalence of the bivariate polynomial matrices. In this paper, the equivalence problems on several classes of bivariate polynomial matrices are investigated. Some new results on the equivalence of these matrices are obtained. These results are useful for reducing two-dimensional systems.

1. Introduction

Multidimensional (nD) systems, especially two-dimensional (2D) systems, are widely used in the field of circuits, image, signal processing, control systems, etc. [1–5, 12]. And the theory of 2D systems has received increasing attention, among which the reduction of 2D systems is an important research content. Usually, a given system is desirable to reduce to an equivalent system with fewer equations or unknowns (named the equivalence of 2D systems). In this way, the characteristics of the original system can be studied in a better and simpler way. A 2D system can be represented with two types of dynamical elements, so 2D systems are often described by bivariate polynomial matrices. Hence, the equivalence problems of 2D systems are usually transformed into the equivalence problems of polynomial matrices [6–11].

The equivalence problem of univariate polynomial matrices was solved by Rosenbrock in 1970 [6]. Using the Euclidean division property of the univariate polynomial ring, he proved that every univariate polynomial matrix in this ring is equivalent to the Smith form. The equivalence problem of the bivariate polynomial matrix is more complex. Lee and Zak gave an example of bivariate polynomial matrix , which is not equivalent to the Smith form [8]. Note that none of the bivariate polynomial rings is a Euclidean ring, and the Euclidean division property does not hold. So, many researchers study the equivalence of some special types of bivariate polynomial matrices. In 1986, Frost and Boudellioua proved that a full row rank bivariate polynomial matrix is equivalent to the Smith form if and only if there exists a unimodular column vector such that has a right inverse [9]. In 2012, Boudellioua wrote an algorithm to enable the decomposition and equivalence of some bivariate polynomial matrices to be realized in Maple [10]. In 2019, Li et al. presented some criteria on an bivariate polynomial matrix which is equivalent to the Smith form diag with being an irreducible polynomial [11]. There are also some results on the equivalence of multivariate polynomial matrices in special cases [13, 14, 16, 18, 19], among which when an polynomial matrix which is equivalent to the Smith form diag is investigated, such as [13], [14], and [19], diag is a very important kind of matrices in the equivalence of multidimensional systems [9, 15, 17].

In this paper, denotes a bivariate polynomial ring with being a field, and we consider arbitrary polynomial in as a polynomial in with coefficient in , written as , where . Note that the coefficient ring is a Euclidean ring, and combined with the Euclidean division property of , we will investigate some classes of bivariate polynomial matrices with their entries in . The following three problems are also considered.

Problem 1. Let , and , where is irreducible and is a positive integer. When is equivalent to

Problem 2. Let with , be irreducible, and be positive integers. When is equivalent to

Problem 3. Let with , where denotes the greatest common divisor of the minor of , is irreducible, and are positive integers. When is equivalent to its Smith form?

2. Preliminaries

In the following, is an arbitrary field, is the univariate polynomial ring in variable with coefficients in , is the bivariate polynomial ring in variables whose coefficients are in , is the algebraic closed field of , is the zero matrix, and is the identity matrix. For , will be the greatest common divisor (g.c.d) of the minors of , . Set , where , is irreducible, and , where denotes . For convenience, the argument is omitted whenever its omission does not cause confusion throughout this paper.

Definition 1. (see [16]). Let with rank and be a polynomial defined as follows:where , is the g.c.d of the minors of , and satisfiesThen, the Smith form of is given by

Definition 2. (see [16]). Let with full row rank; is said to be zero left prime if the minors of have no common zero in .

Definition 3. Let ; then, we say to be a unimodular matrix if the determinant of is a unit of .

Definition 4. Let ; is said to be equivalent to if there are unimodular matrices and such that

Lemma 1 (see [16]). Suppose , ; if is equivalent to , then , for .

Lemma 2 (see [16]). Let , , , and . If the minors of have no common zero in , then the minors of have no common zero in for .

3. Main Results

In this section, we investigate the three problems presented in Section 1 and give the main results of this paper.

In the following, for , we consider it as an element in , written as , where . If is irreducible, then denotes and or 0, where , . Hence, and are relatively prime in ; by Euclidean algorithm, there are such that , i.e., . Then, , where is monic. In other words, can be reduced to a monic polynomial by .

Denote

First, we investigate Problem 1.

Theorem 1. Let with , where is irreducible. If , then is equivalent to the Smith form .

Proof. If rows of are zero vectors mod , that is, rows of are zero vectors, we obtain thatand then , where is unimodular; by computing, , so is unimodular. Hence, is equivalent to the Smith form .
If has no rows which are the zero vectors mod , then has rows of zero vectors mod ; in other words, has rows of zero vectors, . We premultiply and postmultiply by unimodular matrices and such thatwhere , , or . Let , where , and . Note that and are relatively prime, so we can find such that ; then, . We have , where is monic. There are such that , where or . Therefore,where or .
LetThen,where or . If some of are not 0, , do some row transformations to the matrix such that the nonzero polynomial of the least degree in among its first column is at position (1, 1). Repeating the previous steps, we obtain thatwhere is a unimodular matrix, the last rows of are zero vectors, and .
If rows of are zero vectors mod , we can find two unimodular matrices such thatwhere .
Let and ; we have thatwhere and . Then, , where and are unimodular matrices, so is equivalent to the Smith form .
If has no rows of zero vectors mod , then has rows of zero vectors, . Imitating the previous procedure to , there are two unimodular matrices such thatwhere and the last rows of are zero vectors.
Let and ; then,Repeating the procedure above successively, we obtain a series of . If there is some which contains rows of zero vectors mod , then the conclusion is straightforward. Otherwise, has no rows of zero vectors mod .
Then, we consider the case that has no rows of zero vectors mod , . In this case, , and it has no rows of zero vectors mod ; there are unimodular matrices such thatwhere .
Let and ; then,Letcombined with ; we have that and . Considering the minors of , we see that for all . Since , , . Hence, . Hence,where . Note that and are unimodular matrices, ; then, we obtain that and is a unimodular matrix. Let and ; then, with being unimodular matrices. Therefore, is equivalent to the Smith form .

Remark 1. Theorem 1 provides a positive answer to Problem 1.

Corollary 1. Let and be an irreducible polynomial. If and , then can be factorized as , where and is unimodular.

Theorem 2. Let and be an irreducible polynomial. Suppose , and are positive integers. If , then can be factorized aswhere is defined as above and are unimodular matrices, .

Proof. When , by Theorem 1, , where are unimodular matrices, so the conclusion is true.
When , since is a factor of , according to Corollary 1, we see that can be factorized as , where is a unimodular matrix.
Then, we consider . For , according to Lemma 1, we obtainSo,Note that ; then, . According to Corollary 1, can be factorized as . Then,Imitating the procedure above successively, we can obtain thatwhere are unimodular matrices, .

Lemma 3. Let be defined as above and be unimodular. Suppose . If the minors of have no common zero in , then is equivalent to .

Proof. Letwhere , , , and .
Note thatThen,By computing, . Because is unimodular and the minors of have no common zero,Since is ZLP, the minors of have no common zero. Set the minors of to be , where . Then, the minors of are , where .
We prove that is ZLP.
Suppose the minors of have a common zero ; then, is a common zero of and . Note thatThen, the minors of have common zero . This is a contradiction. So, the minors of have no common zero. Hence, is ZLP. By the Quillen–Suslin theorem, we can find a unimodular matrix that satisfiesThen,where and . Letsuch thatLetWe know is equivalent to .
According to Lemma 1, . Hence,Notice thatWe have that divides every element of , soHence, , and . Note that , so ; then, , where and are unimodular matrices. Thus, is equivalent to .

Theorem 3. Let and be an irreducible polynomial. If and , where and are positive integers, then is equivalent to the Smith formif and only if the minors of have no common zero in .

Proof. Sufficiency: by Theorem 2, we have thatwhere are unimodular matrices, .
We first prove that is equivalent to . From the minors of which have no common zero in , we know that the minors of have no common zero in by combining with Lemma 2. Then, according to Lemma 3, is equivalent to .
Repeating the procedure above, we obtain that is equivalent to the matrix , so there exist such thatwhere are unimodular matrices. Hence, we have thatwhere and are unimodular matrices. Therefore, is equivalent to the Smith form .
Necessity: since is equivalent to , there exist unimodular matrices and such that . For the minors of which have no common zero in , the minors of have no common zero in by Lemma 2.

Remark 2. According to Theorem 3, Problem 2 is solved, and a criterion for discriminating this class of bivariate polynomial matrices to be equivalent to the Smith form is also presented.

Theorem 4. Let be of full row rank and be irreducible. Suppose and , where and are positive integers. Then, is equivalent to the Smith formif and only if the minors of have no common zero in .

Proof. Sufficiency: by Theorem 3.3 in [17], there are and satisfying , where and is ZLP. Then, the minors of have no common zero by using Lemma 2. Combined with and Theorem 2, there are unimodular matrices , such that . Then,We know that is also ZLP ( is unimodular). According to the Quillen-Suslin theorem, there exists an unimodular matrix which satisfies that . Then,According to the minors of which have no common zero and , we see that is the Smith form of . Note that and are invertible matrices, so is equivalent to the Smith form .
Necessity: since is equivalent to the Smith form , it is easy to see that the minors of have no common zero in . By Lemma 2, we have that the minors of have no common zero in .

Remark 3. A positive answer to Problem 3 is presented in Theorem 4. And the equivalence of a rectangle matrix and its Smith form is considered, which makes the result more general.

4. An Example

In this section, we use an example to illustrate our results and methods.

Example 1. Consider a 2D polynomial matrix of :where

By computing, , , , and . We know that the minors of have no common zero in . Combining is irreducible and ; by Theorem 3, we have that is equivalent to the Smith form

Let denote mod ; then,and no row of is zero vector mod . Note that the nonzero polynomial of the least degree in among the first column is , so we postmultiply by a unimodular matrix such that can be changed to the position (1, 1). Then,

We consider the element ; is the leading coefficient of . We know that and are relatively prime in , so we can find , such that ; then, . So, ; that is, can be reduced to a monic polynomial by mod .

Then, we reduce other elements in the first column.

For the element , we can find and such that . In reality, .

And for the element , we can find and such that

Actually, .

Let

We havesowhere

Let ; then,

Now, consider , for none of its rows are zero vectors mod ; repeating the steps above, we can obtain thatwhere

Hence,where

So,

Combiningwe have that

Consider ; combined with Lemma 3, we obtain

Let

Then,

So,and by computing, and are unimodular matrices. Therefore, is equivalent to .

5. Conclusions

In this paper, we have investigated the reduction of several kinds of bivariate polynomial matrices in , where is an arbitrary field. Some new results on these matrices to be reduced to their Smith forms are presented. Furthermore, the conditions of these results are easily verified. An example is given to illustrate our method in the end of the article. All of these are useful for reducing 2D systems.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11871207 and 11971161).

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Copyright © 2021 Dongmei Li et al. 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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