On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains

Liu, Jinwang; Wu, Tao; Li, Dongmei; Guan, Jiancheng

doi:https://doi.org/10.1155/2020/1684893

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries 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 1684893 | https://doi.org/10.1155/2020/1684893

On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains

Jinwang Liu,¹Tao Wu,¹Dongmei Li,¹and Jiancheng Guan¹

Guest Editor: Hou-Sheng Su

Received17 Feb 2020

Accepted26 Mar 2020

Published22 Apr 2020

Abstract

In this paper, zero prime factorizations for matrices over a unique factorization domain are studied. We prove that zero prime factorizations for a class of matrices exist. Also, we give an algorithm to directly compute zero left prime factorizations for this class of matrices.

1. Introduction

Multidimensional linear systems theory has a wide range of applications in circuits, systems, control of networked systems, signal processing, and other areas (see, e.g., [1, 2]). Multivariate polynomial matrix theory is a well-established tool for these systems, since many problems in the analysis and synthesis of control systems can be well solved using multivariate polynomial matrix techniques [1–3].

In recent years, -D polynomial matrix factorizations have been widely studied [4–10]. In [11, 12], the zero left prime factorization problem was raised. This problem has been solved in [4–6]. The minor left prime factorization problem has been solved in [7, 10]. In the algorithms given in [7, 10], a fitting ideal of some module over the multivariate (-D) polynomial ring needs to be computed. It is a little complicated.

It is well known that a multivariate polynomial ring over a field is a unique factorization domain. Then, the following problem is interesting.

Problem 1. How to decide if a matrix with full row rank over a unique factorization domain has a zero left prime factorization?
In this paper, we will give a partial solution to this problem.

2. Preliminaries

Let be a unique factorization domain. The set of all matrices with entries from is denoted by . Let . We denote the greatest common divisor of all minors of by . Let be a submatrix of . By deleting from , we get a submatrix of . This submatrix is denoted by .

Let . denotes the adjoint matrix of . denotes the th algebraic cofactor of .

Definition 1. Let , and let be a submatrix of . A minor of consisting of columns from and one column from is said to be a related minor of .
The following definition is from the multidimensional systems theory [13].

Definition 2. Let be of full row rank. Then, is said to be zero left prime (ZLP) if the minors of generate the unit ideal . Suppose has a factorization , where . If is ZLP, then this factorization is said to be a zero left prime factorization.

3. Main Results

First, we need a lemma.

Lemma 1. Let , where . Then, the elements of are just all related minors of (up to a sign).

Proof. Let . Let . Then,by Laplace Theorem. Thus, is a related minor of (up to a sign). It is clear that they are just all related minors of (up to a sign).
Now, we prove the main theorem of this paper.

Theorem 1. Let . If there exists an submatrix of such that is a common factor of all related minors of , then there exists such that and is ZLP; i.e., has a ZLP factorization.

Proof. We can change the order of the columns of such that the submatrix consists of the left columns of . Thus, there exists an invertible matrix such that , where . Since is a common factor of all related minors of , by Lemma 1, we have . LetThen, . We haveThen,Let . Then, . Since consists of the upper rows of invertible matrix , we have is ZLP.

Corollary 1. Let . If there exists an submatrix of such that is a common factor of all related minors of , then .

Proof. Clearly, . By Theorem 1, there exists such that . By Cauchy–Binet formula, we have . Therefore, .

Corollary 2. Let . If there exists an submatrix of such that is a common factor of all related minors of , then is equivalent to .

Proof. By Theorem 1, there exists such that and is ZLP. By Quillen–Suslin theorem, there exists such that is an invertible matrix. Since , we have being equivalent to .
Now, let . Suppose there exists an submatrix of such that . We can give an algorithm to directly compute the ZLP factorization of .

Algorithm 1. (i)Compute all minors of and .(ii)Find an submatrix of such that .(iii)Compute invertible matrix such that .(iv)Let and . Then, .Now, we give an example to illustrate this algorithm.

Example 1. Let , and letThen, . LetThen, is a submatrix of and . LetThen, , whereThus, . LetThen,LetThen, .

Data Availability

No data were used to support this study.

Conflicts of Interest

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

Acknowledgments

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

References

N. K. Bose, Applied Multidimensional Systems Theory, Van Nostrand Reinhold, New York, NY, USA, 1982.
N. K. Bose, B. Buchberger, and J. P. Guiver, Multidimensional Systems Theory and Applications, Kluwer, Dordrecht, The Netherlands, 2003.
O. Bachelier and T. Cluzeau, “Digression on the equivalence between linear 2D discrete repetitive processes and roesser models,” in Proceedings of the Digression on the Equivalence Between Linear 2D Discrete Repetitive Processes and Roesser Models, pp. 1–6, Zielona Gora, Poland, September 2017.
View at: Publisher Site | Google Scholar
J. F. Pommaret, “Solving bose conjecture on linear multidimensional systems,” in Proceedings of the 2001 European Control Conference (ECC), pp. 1853–1855, Porto, Portugal, September 2001.
View at: Publisher Site | Google Scholar
V. Srinivas, “A generalized Serre problem,” Journal of Algebra, vol. 278, no. 2, pp. 621–627, 2004.
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
M. Wang and C. P. Kwong, “On multivariate polynomial matrix factorization problems,” Mathematics of Control, Signals, and Systems, vol. 17, no. 4, pp. 297–311, 2005.
View at: Publisher Site | Google Scholar
M. Wang, “On factor prime factorizations for -D polynomial matrices,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 6, pp. 1398–1405, 2007.
View at: Publisher Site | Google Scholar
J. Liu and M. Wang, “Further remarks on multivariate polynomial matrix factorizations,” Linear Algebra and its Applications, vol. 465, pp. 204–213, 2015.
View at: Publisher Site | Google Scholar
J. Guan, W. Li, and B. Ouyang, “On minor prime factorizations for multivariate polynomial matrices,” Multidimensional Systems and Signal Processing, vol. 30, no. 1, pp. 493–502, 2019.
View at: Publisher Site | Google Scholar
Z. Lin, “Notes on -D polynomial matrix factorization,” Multidimensional Systems and Signal Processing, vol. 10, no. 4, pp. 379–393, 1999.
View at: Publisher Site | Google Scholar
Z. Lin and N. K. Bose, “A generalization of Serre’s conjecture and some related issues,” Linear Algebra and its Applications, vol. 338, no. 1–3, pp. 125–138, 2001.
View at: Publisher Site | Google Scholar
D. C. Youla and G. Gnavi, “Notes on -dimensional system theory,” IEEE Transactions on Circuits and Systems, vol. 26, no. 2, pp. 105–111, 1979.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Jinwang Liu 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

278

Downloads

565

Citations