Elsevier

Automatica

Volume 101, March 2019, Pages 207-213
Automatica

Brief paper
Constrained two-sided coupled Sylvester-type quaternion matrix equations

https://doi.org/10.1016/j.automatica.2018.12.001Get rights and content

Abstract

In this paper, we consider the solvability of a system of constrained two-sided coupled generalized Sylvester quaternion matrix equations. Some necessary and sufficient conditions for the existence of solutions to this system are derived in terms of the ranks and the generalized inverses of the coefficient matrices. Moreover, we find an expression of the general solution to this system when the solvability conditions are satisfied. Finally, a numerical example is presented to illustrate our results.

Introduction

Sylvester-type matrix equations play an important role in control systems (Syrmos & Lewis, 1993), image processing (Calvetti & Reichel, 1996), statistics and probability (Barbour & Utev, 1998), neural network (Zhang, Jiang, & Wang, 2002), eigenvalue assignment problems (Brahma & Datta, 2009), graph theory (Dmytryshyn & Kågström, 2015), and so on.

Quaternions and quaternion matrices have been used in many fields such as control theory, signal and color image processing, orbital mechanics and computer science. (e.g. Bihan and Mars, 2004, He et al., 2016, He et al., 2019, Jia et al., 2018, Leo and Scolarici, 2000, Took and Mandic, 2011). Sylvester-type matrix equations over the quaternion algebra have attracted the attention of many researchers (e.g., Futorny et al., 2016, He et al., 2017, Wang, 2004). For instance, Rodman considered the standard one-sided Sylvester quaternion matrix equation (page 117, Rodman, 2014). Wang et al. considered some systems of one-sided coupled Sylvester-type quaternion matrix equations (He and Wang, 2017, He et al., 2018, Wang and He, 2013, Wang and He, 2014, Wang et al., 2016).

The numerical solution to the system of two-sided coupled Sylvester-type matrix equations was first considered by Byers and Rhee (1995). Jonsson and Kågström (2002) gave algorithms to solve some triangular systems of two-sided generalized Sylvester equations. Hajarian (2015) gave an iterative method for solving the system of two-sided coupled Sylvester-type complex matrix equations AkXkBk+CkYkDk=Mk,EkXk+1Fk+GkYkHk=Nk.Note that most of the research work related to two-sided coupled Sylvester-type matrix equations build upon existing works for the complex matrix and numerical solution cases. However, so far there has been little information on the solvability conditions and general solution to two-sided coupled Sylvester-type matrix equations over the quaternion algebra. Due to the noncommutativity, one cannot directly extend various results on real or complex numbers to quaternions. Motivated by the wide applications of Sylvester-type quaternion matrix equations and in order to improve the theoretical development of two-sided Sylvester-type equations, we in this paper consider the solvability conditions and general solution to a system of constrained two-sided coupled Sylvester-type quaternion matrix equations. More specifically, A1X=C1,A2Y=C2,A3Z=C3,XB1=D1,YB2=D2,ZB3=D3,A4XB4+C4YD4=P,A5ZB5+C5YD5=Q,where Aj,Bj,Cj,Dj,P,Q,(j=1,,5) are given quaternion matrices, and X,Y,Z are unknowns.

The remainder of the paper is organized as follows. In Section 2, we discuss the special systems of quaternion matrix equations. In Section 3, we give some necessary and sufficient conditions for the existence of a solution to the system (1). We also present the general solution to the system (1) when it is solvable. A numerical example is presented to illustrate that our results are feasible. Finally, a brief conclusion is given in Section 4.

Throughout this paper, let R and Hm×n stand, respectively, for the real number field and the set of all m×n matrices over the quaternion algebra H={a0+a1i+a2j+a3k|i2=j2=k2=ijk=1,a0,a1,a2,a3R}. For more definitions and properties of quaternions, we refer the reader to the recent book (Rodman, 2014). The symbols r(A) and A stand for the rank of a given quaternion matrix A and the conjugate transpose of A, respectively. The identity matrix and zero matrix with appropriate sizes are denoted by I and 0, respectively. The Moore–Penrose inverse A of a quaternion matrix A, is defined to be the unique matrix A, such that (i)AAA=A,(ii)AAA=A,(iii)(AA)=AA,(iv)(AA)=AA. Furthermore, LA and RA stand for the projectors LA=IAA and RA=IAA induced by A, respectively.

Section snippets

Preliminaries

In this section, we consider some quaternion matrix equations, which will be used in proving the main result. The following lemma gives the solvability condition and general solution to the system of quaternion matrix equations A1X=C1,XB1=D1,where A1,B1,C1 and D1 are given quaternion matrices.

Lemma 2.1

Wang, van der Woude, & Chang, 2009

The system (2) is consistent if and only if RA1C1=0,D1LB1=0,A1D1=C1B1. In this case, the general solution to (2) can be expressed asX=A1C1+LA1D1B1+LA1X1RB1, whereX1 is an arbitrary matrix over H with

Some solvability conditions and the general solution to (1)

In this section, we give some necessary and sufficient conditions for the existence of a solution to the system (1). We also present the expression of the general solution to the system (1) when it is solvable. In addition, a numerical example is presented to illustrate the main result.

Now we give the fundamental theorem of this paper. For simplicity, put A11=A4LA1,B11=RB1B4,C11=C4LA2,D11=RB2D4,A22=A5LA3,B22=RB3B5,C22=C5LA2,D22=RB2D5,E11=PA4(A1C1+LA1D1B1)B4C4(A2C2+LA2D2B2)D4,E22=QA5(A3C3

Conclusion

We have derived some necessary and sufficient conditions for the existence of a solution (X,Y,Z) to the system of constrained two-sided coupled generalized Sylvester quaternion matrix equations (1). We have also given the general solution to the system (1) when it is solvable. The results in this paper are also valid over the real number field and complex number field as special cases.

Qing-Wen Wang was born in Shandong Province, China, in 1964. He achieved his Ph.D. degree from the University of Science and Technology of China. Since January 2004, he has been a Full Professor and the head of the Department of Mathematics at Shanghai University. His research interests include matrix theory, linear algebra and its applications in systems and control, operator algebra, numerical linear algebra, and quantum computing. He has authored or co-authored more than 100 papers in

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    Qing-Wen Wang was born in Shandong Province, China, in 1964. He achieved his Ph.D. degree from the University of Science and Technology of China. Since January 2004, he has been a Full Professor and the head of the Department of Mathematics at Shanghai University. His research interests include matrix theory, linear algebra and its applications in systems and control, operator algebra, numerical linear algebra, and quantum computing. He has authored or co-authored more than 100 papers in refereed journals. He has also been active in his service as the Editor-in-Chief and editors for several refereed journals.

    Zhuo-Heng He was born in Zhejiang Province, China, in 1987. He received his Bachelor degree, Master Degree and Ph.D. degree from the Department of Mathematics at Shanghai University, in 2010, 2013 and 2016, respectively. He was a joint Ph.D. student in the Department of Electrical Engineering (ESAT), KU Leuven, Belgium, from 2014 to 2015. From 2016 to 2018, he was a visiting Assistant Professor of the Department of Mathematics and Statistics at Auburn University. Since 2018, he has been an Assistant Professor of the Department of Mathematics at Shanghai University. His research interests include matrix theory, linear algebra and its applications in systems and control, numerical linear algebra.

    Yang Zhang was born in Henan Province, China, in 1965. He received his Ph.D. degree from the University of Western Ontario, Canada. Since July of 2008, he has been an Associate Professor in Department of Mathematics at the University of Manitoba, Canada. His recent research interest is Symbolic Computation, in particular, efficient algorithms for matrices over noncommutative rings, factoring skew polynomials, Gröbner bases and applications. His research is supported by The Natural Sciences and Engineering Research Council of Canada (NSERC).

    This research is supported by the grants from the National Natural Science Foundation of China (11571220) and (11801354), the Macao Science and Technology Development Fund (185/2017/A3), and the Natural Sciences and Engineering Research Council of Canada (NSERC) (RGPIN 312386-2015). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Petersen.

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