On Serre Reduction of Multidimensional Systems

Li, Dongmei; Liu, Jinwang; Zheng, Licui

doi:https://doi.org/10.1155/2020/7435237

Mathematical Problems in Engineering

On this page

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

Special Issue

Control of Networked Systems with Engineering Applications 2020

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 7435237 | https://doi.org/10.1155/2020/7435237

On Serre Reduction of Multidimensional Systems

Dongmei Li,¹Jinwang Liu,¹and Licui Zheng¹

Guest Editor: Hou-Sheng Su

Received17 Feb 2020

Accepted17 Mar 2020

Published11 Apr 2020

Abstract

Serre reduction of a system plays a key role in the theory of Multidimensional systems, which has a close connection with Serre reduction of polynomial matrices. In this paper, we investigate the Serre reduction problem for two kinds of D polynomial matrices. Some new necessary and sufficient conditions about reducing these matrices to their Smith normal forms are obtained. These conditions can be easily checked by existing Gröbner basis algorithms of polynomial ideals.

1. Introduction

Multidimensional (D) systems arise naturally in several fields of control, circuits, signals, and network synthesis (see [1–5]). Serre reduction problem aims at simplifying D systems to some equivalent forms with fewer equations in fewer unknowns, and rewriting these systems such that the key information on them [6], which does not show obviously in the original forms, can be derived from the new equivalent forms more easily. In general, this involves Serre reduction of multivariate (D) polynomial matrices to some simple equivalent forms, especially their Smith normal forms. Serre reduction can help to investigate the structural of D polynomial matrices, which has wide applications in image, complex networked system, and other areas (see [6–11]).

In the theory of D systems, the Serre reduction for a polynomial matrix to its Smith normal form plays a critical role. It preserves relevant system properties from the system theory point of view. The reduced matrix has good properties and corresponds to a system which consists of much less equations in unknowns than its original form. For single variable polynomial matrices, the problem for reducing the matrix to its Smith normal form has been well solved. Frost and Storey presented an example for 2D polynomial matrices which cannot be reduced to its Smith normal form [12]. For D case, since this reduction problem is equivalent to a highly difficult problem, the isomorphism of two finitely presented modules, the criteria determining whether arbitrary D polynomial matrix can be Serre reduced to its Smith normal form has not been presented so far. The reduction for several special classes of polynomial matrices to their Smith normal forms has been investigated. For the polynomial ring , Lee and Zak [13] presented a necessary condition for a class of 2D polynomial matrices on reducing them to the Smith normal forms. Lin et al. [14] proposed a sufficient and necessary condition, the existence of a special polynomial vector, under which a kind of D polynomial matrices can be reduced to the Smith normal forms. Lin et al. [14] generalized this result to the case when , . Furthermore, they showed that a class of D polynomial matrices can be reduced to its Smith normal form diag for det . Boudellioua [15] gave some similar conditions using module theoretic approach. Li et al. [16] proposed a sufficient and necessary condition under which can be reduced to its Smith normal form diag for and det , .

The results that we mentioned above mainly study polynomial matrices whose reduced matrices correspond to systems which contains only one equation in one unknown with form diag , where is the polynomial matrix corresponds to the given D system. In practical applications, the reduced matrix corresponds to a given system which is a more general form with more than one equation in unknowns, such as, diag . In this paper, we will investigate these systems and the reduced matrices. The following questions are considered.

Problem 1. Let and , where is a positive integer. When is reduced to its Smith normal form diag ?

Problem 2. Let has full rank and , where is the greatest common divisor (g.c.d.) of all the minors of and is a positive integer. Is there a tractable criterion for to be reduced to its Smith normal form?
This paper is organized as follows. We present some basic preliminary knowledge and main results for reducing several kinds of matrices to their Smith normal forms in Section 2. An example is established to illustrate our results and the constructive computational method in Section 3.

2. Preliminaries and Results

In the following, denotes the polynomial ring in variables with coefficients in the field , denotes the algebraic closed field of , denotes the set of matrices with entries in , denotes the zero matrix, and denotes the identity matrix. The argument (z) is omitted whenever its omission does not cause confusion, for example, we denote by for simplicity.

For , we use to denote the greatest common divisor of all the minors of .

Definition 1 (see [17]). Suppose be of full row (column) rank, we say is zero left prime (zero right prime) if all the minors of have no common zero or generate the unit ideal .
If is a zero left prime (zero right prime) matrix, we simply say that is ZLP (ZRP). By Quillen–Suslin theorem [18], we see that is ZLP iff there exists an invertible matrix such that . Quillen–Suslin theorem states that finite generated projective -modulo is free, and it is equivalent to saying that any ZLP (ZRP) matrix on can be completed to an invertible (unimodular) square matrix.

Definition 2 (see [17]). Let , , the Smith normal form of is defined aswhere is the rank of , , is the g.c.d. of the minors of , and satisfies the divisibility property:

Definition 3 (see [17]). Let and be two matrices in , then and are said to be equivalent (or can be reduced to if there exist two invertible (unimodular) matrices and such that ).
First, we give the following several well-known results about the factorization of D polynomial matrices [10, 17, 19, 20].

Lemma 1 (see Hilbert zeros Theorem [17, 20]). Suppose , then , have no common zero in if and only if the ideal generated by is the unit ideal .

Lemma 2 (see [19]). Suppose is of full column rank and the reduced minors of generate , then admits a zero prime factorization with , , and , and is a ZRP matrix.

Remark 1. Suppose has full row rank and the reduced minors of generate , then admits the factorization as with , , and det . Hence, is a ZLP matrix.

Lemma 3 (see [17, 20]). Suppose has normal rank , if the reduced minors of generate , then there exists a ZLP matrix such that .

Lemma 4 (see [17]). Let and . Suppose that , then is a divisor of .

In what follows, for a polynomial matrix , we use to denote an submatrix of constituted by the rows and columns of .

Lemma 5. Suppose with . If all the minors of have no common zero, for some , then all the minors of have no common zero.

Proof. For any , sinceaccording to Cauchy–Binet formula, we obtainwhere and are minors of and , respectively. Suppose that all the minors of have a common zero point . So, the minors of have a common zero point . This contradicts the assumption. Therefore, all the minors of have no common zero.

Lemma 6. Suppose and can be reduced to , then for , .

Proof. Since can be reduced to , by Definition 3, there are invertible matrices and such that . Setting , according to Cauchy–Binet formula, we have thatwhere is a minors of , is a minors of , and so . Note that , similar to the procedure above, we have that . Hence, .
In the following, we denote by and by simply. We can obtain the following result, which gives a positive answer to Problem 1.

Theorem 1. Suppose has a full row rank and det , where , is a positive integer. If all the minors of have no common zero and , then can be reduced to its Smith form:with det .

Proof. We first prove that rank is . Since , it follows that rank . Because minors of have no common zero, then . Combined with Lemma 5, we obtain that , and rank . Let , be the minors of and , respectively. It is straightforward that is a divisor of . Since have no common zero, then , also have no common zero. So, , also have no common zero, i.e., the minors of have no common zero.
We then show that can be reduced to its Smith normal form. According to Lemma 3, there exists a ZLP polynomial matrix which satisfies . By Quillen–Suslin Theorem, we can complete into an unimodular matrix , that is, is the last rows of . We have that the last rows of the matrix are the zero polynomials. According to Lemma 4, there is the common divisor in the last rows of , i.e.,where . Setting , then det . Combining with , , and det , we obtain that the Smith normal form of is . LetThen, det and . Note that are unimodular matrices, so can be reduced to its Smith normal form .
DenoteIn order to investigate Problem 2, we consider the following simple case first.

Theorem 2. Suppose has full row rank with , where is a positive integer. If all the minors of have no common zero and , then can be reduced to its Smith normal form.

Proof. Since all the minors of have no common zero, by Lemma 1, then all the minors of generate . Combined with , we have that the Smith normal form of is the matrix:According to Remark 1, there exist with , such that , and is ZLP. Combining with Lemma 5, we have that the minors of have no common zero and . From Theorem 1, there exist two unimodular polynomial matrices which satisfy . It is straightforward that is also ZLP. According to Quillen–Suslin theorem, there is an invertible matrix such that . Hence,Thus, can be reduced to its Smith normal form .

Lemma 7. Suppose is a unimodular matrix. If all the minors of have no common zero, then the matrix can be reduced to the matrix .

Proof. Setting andwhere , , , and , we obtain thatNote that is ZLP (since an invertible matrix), and according to Quillen–Suslin theorem, there exists an invertible matrix such that . Settingwe partition and aswhere and .
We obtain thatSettingwe have thatSince is ZLP, according to Quillen–Suslin theorem, there exists an invertible matrix such that . Settingwe see thatObviously, and are invertible. So, can be reduced to

Theorem 3. Suppose with det . If all the minors of have no common zero and , where are positive integers. Then, can be reduced to its Smith normal form:

Proof. When , by Theorem 1, the conclusion is correct.
When , since , it follows that rank . All the minors of have no common zero, and doing the same as that of the proof of Theorem 1, we obtain that rank andwhere is a unimodular matrix and . Hence,We know that from the equation above. According to Lemma 5, all the minors of have also no common zero. If , combined with is a positive integer, then ; imitating the procedure above, we obtain thatThen,Using Lemma 5 again, all the minors of also have no common zero. Combined with Lemma 7, two invertible matrices exist and satisfy . Therefore,Note that and , then . Furthermore, from Lemma 5, we see that the minors of also have no common zero.
If , repeating the previous process, we obtain thatNote that and , we see . Hence, and are invertible matrices. So, can be reduced to its Smith normal form .

Theorem 4. Suppose () has full row rank with and , where and are positive integers. Then, can be reduced to its Smith normal form if and only if all the minors of have no common zero.

Proof (sufficiency). Since all the minors of have no common zero and , then the Smith normal form of isAccording to Remark 1, there exist with , satisfy , and is ZLP. Combining with Lemma 5, we have that the minors of have no common zero and . From Theorem 3, there are invertible matrices which satisfy . It is straightforward that is also a ZLP matrix. From Quillen–Suslin theorem, we have that there is an invertible matrix which satisfy . Hence,So, can be reduced to its Smith normal form .

Necessity. Assume is the Smith normal form of . It is obvious that all the minors of have no common zero and . Since can be reduced to , by Lemmas 5 and 6, we have that all the minors of have no common zero and .

Remark 2. Let be some nonzero polynomials in ; by Lemma 1, we know a necessary and sufficient condition of have no common zero. is the reduced Gröbner basis of the ideal , which is generated by , and contains a unit of . By Theorems 1–4, we can check whether such a kind of D polynomial matrices can be reduced to its Smith normal form by using the existing Gröbner basis algorithms. So, the conditions of Theorems 1–4 can be verified easily.

3. An Example

In this section, we present an example to explain our method on how to reduce the polynomial matrix we considered to its Smith normal form.

Example 1. We consider the following polynomial matrix:with . Then, we will prove that can be reduced to its Smith normal form.

Step 1. By computing, and the reduced Gröbner basis of the ideal, which is generated by the minors of , is . Combined with Remark 2 and Lemma 1, we have that the minors of have no common zero. Let , and then according to Theorem 3, we have that is can be reduced to its Smith normal form:where .

Step 2. Substituting for in , we haveThen, we construct the following invertible matrix:such thatComputing by singular,

Step 3. Supposerepeating Step 2 to , we haveHence,

Step 4. Letwe check easily that is an invertible matrix.

Step 5. By Lemma 7, we obtainHence,Thus,where and are invertible matrices.

4. Conclusions

In this paper, we have investigated the Serre reduction problem for two classes of multivariate polynomial matrices with the g.c.d. of maximal order minors as . We have obtained some tractable sufficient and necessary conditions under which such kinds of matrices can be reduced to their Smith forms. These conditions can be checked by using the existing Gröbner basis algorithms to the associated minors of the given matrix. We also give an example to illustrate our method. All of these could provide useful information for reducing D 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).

References

E. Antoniou and S. Vologiannidis, “On the reduction of 2-D polynomial system into first order equivalent models,” Multidimensional Systems and Signal Processing, vol. 31, no. 1, 2020.
View at: Publisher Site | Google Scholar
H. Su, J. Zhang, and X. Chen, “A stochastic sampling mechanism for time-varying formation of multiagent systems with multiple leaders and communication delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 30, no. 12, pp. 3699–3707, 2019.
View at: Publisher Site | Google Scholar
O. Bachelier and T. Cluzeau, “Digression on the equivalence between linear 2d discrete repetitive processes and roesser models,” in Proceedings of the 2017 10th International Workshop on Multidimensional (nD) Systems (nDS), pp. 1–6, Zielona Gora, Poland, September 2017.
View at: Publisher Site | Google Scholar
N. Bose, Applied Multidimensional Systems Theory, Van Nostrand Reinhold, Newyork, NY, USA, 1982.
N. Bose, B. Buchberger, and J. Guiver, Multidimensional Systems Theory and Applications, Kluwer, Dordrecht, The Netherlands, 2003.
M. S. Boudellioua and A. Quadrat, “Serre’s reduction of linear functional systems,” Mathematics in Computer Science, vol. 4, no. 2-3, pp. 289–312, 2010.
View at: Publisher Site | Google Scholar
H. Su, Y. Ye, Y. Qiu, Y. Cao, and M. Z. Q. Chen, “Semi-global output consensus for discrete-time switching networked systems subject to input saturation and external disturbances,” IEEE Transactions on Cybernetics, vol. 49, no. 11, pp. 3934–3945, 2019.
View at: Publisher Site | Google Scholar
C. Xu, Y. Zhao, B. Qin, and H. Zhang, “Adaptive synchronization of coupled harmonic oscillators under switching topology,” Journal of the Franklin Institute, vol. 356, no. 2, pp. 1067–1087, 2019.
View at: Publisher Site | Google Scholar
A. Quadrat, “An introduction to constructive algebraic analysis and its applications,” Les cours du CIRM, vol. 1, no. 2, pp. 281–471, 2010.
View at: Publisher Site | Google Scholar
J. Liu, D. Li, and L. Zheng, “The Lin-Bose problem,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 61, no. 1, pp. 41–43, 2014.
View at: Publisher Site | Google Scholar
C. Xu, H. Su, C. Liu, and G. Zhang, “Robust adaptive synchronization of complex network with bounded disturbances,” Advances in Difference Equations, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar
M. G. Frost and C. Storey, “Equivalence of matrices over R[s, z] : a counter-example,” International Journal of Control, vol. 34, no. 6, pp. 1225-1226, 1981.
View at: Publisher Site | Google Scholar
E. Lee and S. Zak, “Smith normal form over ,” IEEE Transactions on Automatic Control, vol. 28, no. 1, pp. 115–118, 1983.
View at: Publisher Site | Google Scholar
Z. Lin, M. S. Boudellioua, and L. Xu, “On the equivalence and factorization of multivariate polynomial matrices,” in Proceedings of the 2006 IEEE International Symposium on Circuits and Systems, pp. 4911–4914, Island of Kos, Greece, May 2006.
View at: Publisher Site | Google Scholar
M. S. Boudellioua, “Further results on the equivalence to Smith form of multivariate polynomial matrices,” Control and Cybernetics, vol. 42, no. 2, pp. 543–551, 2013.
View at: Google Scholar
D. Li, J. Liu, and L. Zheng, “On the equivalence of multivariate polynomial matrices,” Multidimensional Systems and Signal Processing, vol. 28, no. 1, pp. 225–235, 2017.
View at: Publisher Site | Google Scholar
D. Li, J. Liu, and D. Chu, “The Smith form of a multivariate polynomial matrix over an arbitrary coefficient field,” Linear and Multilinear Algebra, pp. 1–14, 2020.
View at: Publisher Site | Google Scholar
D. Quillen, “Projective modules over polynomial rings,” Inventiones Mathematicae, vol. 36, no. 1, pp. 167–171, 1976.
View at: Publisher Site | Google Scholar
M. Wang and D. Feng, “On Lin-Bose problem,” Linear Algebra and Its Applications, vol. 390, pp. 279–285, 2004.
View at: Publisher Site | Google Scholar
J. Liu, D. Li, and L. Zheng, “Factorization for n-D polynomial matrices over an arbitrary coefficient field,” Journal of Systems Science and Complexity, vol. 33, no. 1, pp. 196–210, 2020.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 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.

PDF Download Citation

Download other formats

Order printed copies

Views

196

Downloads

517

Citations