Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application

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Abstract

In this paper, we establish the maximal and minimal ranks of the solution to the consistent system of quaternion matrix equations A1X=C1,A2X=C2,A3XB3=C3 and A4XB4=C4, which was investigated recently by Wang [Q.W. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 665–675]. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition for the invariance of the rank of the general solution to the system mentioned above is presented. Some previous known results can be regarded as the special cases of this paper.

Introduction

Investigating extreme ranks of matrix expressions has many immediate motivations in matrix analysis. For example, the classical matrix equation A1X1B1+A2X2B2=C is consistent if and only ifminX1,X2rank(C-A1X1B1-A2X2B2)=0;the two consistent matrix equations A1X1B1=C1,A2X2B2=C2, where X1 and X2 have the same size, have a common solution if and only ifminX1,X2rank(X1-X2)=0.In recent years, Tian has investigated extremal ranks of some matrix expressions over a field, and derived many rank formulas and their numerous consequences and applications (see, e.g. [1], [2], [3], [4], [5], [6], [7], [8]).

Solvability and solutions of linear matrix equations have been one of principle topics in matrix analysis and its applications. Researches on extreme ranks of solutions to linear matrix equations has been actively ongoing for more than 30 years. For instance, in 1972, Mitra [9] considered solutions with fixed ranks for the matrix equations AX=B and AXB=C. In 1984, Mitra [10] gave common solutions of minimal rank of the pair of complex matrix equations AX=C, XB=D. In 1987, Uhlig [11] presented the extremal ranks of solutions to the matrix equation AX=B. In 1990, Mitra studied the minimal ranks of common solutions to the pair of matrix equations A1XB1=C1 and A2XB2=C2 over a general field in [12]. In 2003, Tian in [6], [7] has investigated the extremal rank solutions to the complex matrix equation AXB=C and gave some applications. In 2006, Lin and Wang in [13] studied the extreme ranks of solutions to the system of matrix equations A1X=C1,XB2=C2,A3XB3=C3 over an arbitrary division ring, which was investigated in [19] and [20]. Very recently, Wang et al. [14] presented formulas of extreme ranks of a quaternion matrix expression f(X1,X2)=A-A3X1B3-A4X2B4, where X1,X2 are variant quaternion matrices, subject to two consistent systems of quaternion matrix equations A1X1=C1,X1B1=C2 and A2X2=C3,X2B2=C4. In 2005, Wang [15], [16], [17] gave necessary and sufficient conditions for the existence and expression of the general solution to the system of quaternion matrix equationsA1X=C1,A2X=C2,A3XB3=C3,A4XB4=C4and presented its applications. Notice that the linear matrix equations mentioned above are special cases of the system (1.1). As far as the authors know, the extremal ranks of the solution to the systems (1.1) have not been considered so for. In this paper, we mainly consider the extreme ranks of the solution to the system (1.1). In Section 2, we investigate the maximal and minimal ranks of the solution to the system (1.1) under the assumption that (1.1) is consistent over the quaternion field by using some formulas of extreme rank of a matrix expression due to Tian [3], [4], which can be generalized to the quaternion field. Some special cases are also considered in Section 3. As an application, we also establish in Section 4 a necessary and sufficient condition for the invariance of the rank of the general solution to the system (1.1). Some further research problems related this paper are also given in the end of this paper.

Throughout, we denote the real number field by R, the set of all m×n matrices over the quaternion algebraH={a0+a1i+a2j+a3k|i2=j2=k2=ijk=-1,a0,a1,a2,a3R},by Hm×n, the identity matrix with the appropriate size by I, the column right space, the row left space of a matrix A over H by R(A), N(A), respectively, the dimension of R(A) by dimR(A), an inner inverse of a matrix A by A which satisfies AA-A=A, a reflexive inverse of matrix A over H by A+ which satisfies simultaneously AA+A=A and AA+A=A. Moreover, RA and LA stand for the two projectors LA=I-A+A, RA=I-AA+ induced by A, where A+ is any but fixed reflexive inverse of A. Clearly, RA and LA are idempotent and one of its reflexive inverses is itself. By [25], for a quaternion matrix A, dimR(A)=dimN(A). dimR(A) is called the rank of a quaternion matrix A and denoted by r(A).

Section snippets

Extreme ranks of the solution to system (1.1)

We begin with the following Lemmas.

Lemma 2.1

See Lemma 2.2 in [15]

Let A1Hm×n, A2Hs×n, C1Hm×r, C2Hs×r be known and XHn×r unknown, S=A2LA1,G=RSA2. Then the systemA1X=C1,A2X=C2is consistent if and only ifAiAi+Ci=Ci,i=1,2;G(A2+C2-A1+C1)=0.In that case, the general solution of (2.1) can be expressed as the following:X1=A1+C1+LA1S+A2(A2+C2-A1+C1)+LA1LSY,where Y is an arbitrary matrix over H with appropriate dimension.

The following Lemma is due to Tian [18] which can be generalized H.

Lemma 2.2

Consider the consistent system over HA3XB3=C

Extremal ranks of solutions to special cases of system (1.1)

In this section, we consider the special cases of Theorem 2.7.

Corollary 3.1

Consider the consistent matrix equation A3XB3=C3 where A3Hk×p, B3Hq×r, C3Hk×r. Then the maximal and minimal ranks of the solution to the matrix equation aremaxr(X)=minp,q,p+q+r(C3)-r(A3)-r(B3),minr(X)=r(C3).

Proof

In Theorem 2.7, let A1,C1,A2,C2,A4,B4,C4 vanish. Then simplifying (2.11), (2.12) and noticing thatr(C)min{r(A),r(B)},yields (3.1), (3.2). 

Remark 3.1

Corollary 3.1 is Theorem 2.1 of [3].

In Theorem 2.7, let A1, A2 and C1, C2 vanish. Then

The invariance of rank of solution to system of (1.1)

In view of (2.17), the rank of the solution to (1.1) will vary while the variables X1-X4 are changed. As an application of Theorem 2.7, we now investigate the rank invariance of the general solution to system (1.1).

Theorem 4.1

Suppose system (1.1) over H is consistent. Then the rank of the general solution expressed as (2.17) is invariant if and only ifr0C1B4A10C2B4A20C4A4C1B30A1C2B30A2C30A3-rC3A3C1B30C2B300A10A20A4+rC1B30C2B30C300C4B3B4=rC1B30C2B30C30B3B4-rC1B3C2B3C3+rC1B3C1B4C2B3C2B4-rA1A2+porr0C1B4A10C2B

Acknowledgement

The partial work of this paper was finished during the first author’s visit to Department of Mathematics, National University of Singapore.

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This research was supported by the Natural Science Foundation of China (60672160), Shanghai Pujiang Program (06PJ14039), the Development Foundation of Shanghai Educational Committee, and the Special Funds for Major Specialities of Shanghai Education Committee.

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