Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application☆
Introduction
Investigating extreme ranks of matrix expressions has many immediate motivations in matrix analysis. For example, the classical matrix equation is consistent if and only ifthe two consistent matrix equations , where X1 and X2 have the same size, have a common solution if and only ifIn 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 and . In 1984, Mitra [10] gave common solutions of minimal rank of the pair of complex matrix equations , . In 1987, Uhlig [11] presented the extremal ranks of solutions to the matrix equation . In 1990, Mitra studied the minimal ranks of common solutions to the pair of matrix equations and over a general field in [12]. In 2003, Tian in [6], [7] has investigated the extremal rank solutions to the complex matrix equation and gave some applications. In 2006, Lin and Wang in [13] studied the extreme ranks of solutions to the system of matrix equations 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 , where are variant quaternion matrices, subject to two consistent systems of quaternion matrix equations and . 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 equationsand 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 , the set of all matrices over the quaternion algebraby , the identity matrix with the appropriate size by I, the column right space, the row left space of a matrix A over by , , respectively, the dimension of by , an inner inverse of a matrix A by A− which satisfies , a reflexive inverse of matrix A over by A+ which satisfies simultaneously and . Moreover, RA and LA stand for the two projectors , 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, . is called the rank of a quaternion matrix A and denoted by .
Section snippets
Extreme ranks of the solution to system (1.1)
We begin with the following Lemmas. Lemma 2.1 Let , , , be known and unknown, . Then the systemis consistent if and only ifIn that case, the general solution of (2.1) can be expressed as the following:where Y is an arbitrary matrix over with appropriate dimension.See Lemma 2.2 in [15]
The following Lemma is due to Tian [18] which can be generalized . Lemma 2.2 Consider the consistent system over
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 where , , . Then the maximal and minimal ranks of the solution to the matrix equation are Proof In Theorem 2.7, let vanish. Then simplifying (2.11), (2.12) and noticing thatyields (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 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 is consistent. Then the rank of the general solution expressed as (2.17) is invariant if and only ifor
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.