The (P, Q)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations

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Abstract

Let Hm×n denote the set of all m × n matrices over the quaternion algebra H and PHm×m,QHn×n be involutions. We say that AHm×n is (P, Q)-symmetric (or (P, Q)-skewsymmetric) if A = PAQ (or A =  PAQ). We in this paper mainly investigate the (P, Q)-(skew)symmetric maximal and minimal rank solutions to the system of quaternion matrix equations AX = B, XC = D. We present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with (P, Q)-symmetry and (P, Q)-skewsymmetry to the system. The expressions of such solutions to this system are also given when the solvability conditions are satisfied. A numerical example is presented to illustrate our results. The findings of this paper extend some known results in this literature.

Introduction

Throughout this paper, we denote the set of all m × n matrices over the real quaternion algebraH={a0+a1i+a2j+a3k|i2=j2=k2=ijk=-1anda0,a1,a2,a3are real numbers}by Hm×n; the symbols I,A,A,R(A),N(A),dimR(A) stand for the identity matrix with the appropriate size, the transpose conjugate, the Moore–Penrose inverse, the column right space, the left row space of AHm×n, and the dimension of R(A), respectively. For a quaternion matrix A, the two matrices LA and RA stand for the two orthogonal projectors LA = I  AA, RA = I  AA induced by A. By [1], dimR(A)=dimN(A), which is called the rank of A and denoted by r(A).

Into the 20th century, quaternions, discovered by Hamilton in 1843, made further appearance in associative algebras, analysis, topology, and physics. Moreover, quaternion matrices play an important role in computer science, quantum physics, signal and color image processing, and so on (e.g. [2], [3], [4], [5], [6], [7], [8], [9]).

Recall that a reflexive (antireflexive) matrix was defined in [10], [11]: a complex matrix A is reflexive (antireflexive) if A = PAP (A = PAP), where P is a Hermitian involution. In 1998, Chen [12] defined generalized reflexive and antireflexive matrices: a complex matrix A is called generalized reflexive (or antireflexive) if A = PAQ (or A = PAQ) where P, Q are Hermitian involutory matrices. Chen also discussed applications that give rise to these matrices and considered least squares problems involving them. Motivated by generalized reflexive and antireflexive matrices, Trench in 2004 [13] defined (P, Q)-symmetric and (P, Q)-skewsymmetric matrices: a complex matrix A is called (P, Q)-symmetric (or (P, Q)-skewsymmetric) if A = PAQ (or A = PAQ) where P, Q are involutory matrices, i.e. P2 = I, Q2 = I.

In matrix theory and applications, many problems are closely related to the ranks of some matrix expressions with variable entries, and so it is necessary to explicitly characterize the possible ranks of the matrix expressions concerned. The study on the possible ranks of matrix equations can be traced back to the late 1970s (see, e.g. [14], [15], [16], [17], [18]). Recently, the extremal ranks, i.e. maximal and minimal ranks, of some matrix expressions have found many applications in control theory [19], [20], statistics, and economics (see, e.g. [21], [22], [23]).

Consider the system of matrix equationsAX=B,XC=D,which includes the classical matrix equation AX = B and its inverse problem, i.e., the matrix equation XC = D. In 1984, Mitra [15] gave the minimal rank solution to the system of matrix equations over the complex field C. In 2005, Wang [24], [25] investigated the bisymmetric solution to system (1.1). In 2007, using an eigenvalue decomposition of an involution, Qiu et al. [26] gave a simple and eigenvector-free formula of the general reflexive and anti-reflexive solutions to system (1.1) over C. In 2008, Li et al. [27] investigated the generalized reflexive solution to (1.1) over C. Xiao et al. [28] in 2009 considered the symmetric minimal rank solution to the matrix equation AX = B. To our knowledge, so far there has been little information on investigating the (P, Q)-symmetric and (P, Q)-skewsymmetric maximal and minimal rank solutions to system (1.1) over H.

Motivated by the work mentioned above and keeping the interests and applications of quaternion matrices in view, we in this paper mainly consider the (P, Q)-symmetric and (P, Q)-skewsymmetric extreme rank solutions to system (1.1) over H.

We organize this paper as follows. In Section 2, we first establish a representation for a (P, Q)-symmetric (or (P, Q)-skewsymmetric) matrix. Then we give necessary and sufficient conditions for the existence of (P, Q)-symmetric and (P, Q)-skewsymmetric solutions to system (1.1). We also give the expressions of such solutions when the solvability conditions are satisfied. We in Section 3 establish formulas of maximal and minimal ranks of (P, Q)-symmetric and (P, Q)-skewsymmetric solutions to system (1.1), and present the (P, Q)-(skew)symmetric extremal rank solutions to (1.1). We in Section 4 present a numerical example to illustrate the results derived in this paper.

Section snippets

(P, Q)-symmetric and (P, Q)-skewsymmetric solutions to (1.1)

In this section, we first establish the representations of (P, Q)-symmetric and (P, Q)-skewsymmetric quaternion matrices, then investigate the (P, Q)-symmetric and (P, Q)-skewsymmetric solutions to system (1.1).

Due to noncommutation of the multiplication of quaternions, the left eigenvalue and the right one of AHn×n are distinguished, which satisfies the following, respectively, for xH1×n,xA=λx (left), xA =  (right). In this paper, we only use the left eigenvalue (simply say eigenvalue). It can

(P, Q)-(skew)symmetric extremal rank solutions to (1.1)

In this section, we first derive the formulas of the maximal and minimal ranks of (P, Q)-symmetric and (P, Q)-skewsymmetric solutions of (1.1),then present the expressions of (P, Q)-(skew)symmetric maximal and minimal rank solutions to (1.1).

Lemma 3.1

see Lemma 3.2 in [31]

Let A, B and C be arbitrary matrices over H with appropriate sizes. Then the following equalities hold:

  • (a)

    A = (AA)A = A(AA).

  • (b)

    LA = (LA)2 = (LA), RA = (RA)2 = (RA).

  • (c)

    LA(CLA) = (CLA), (RAC)RA = (RAC).

Lemma 3.2

see Lemma 2.3 in [32]

Let AHm×n,BHm×k,CHl×n,DHj×k and EHl×i. Then they satisfy the

A numerical example

In this section, we give a numerical example to illustrate the results of this paper.

Take two involutionsP=1000000.28000.96i000.28-0.96k0000.96k-0.2800-0.96i00-0.28,Q=-0.2800.96j000.280-0.96k-0.96j00.28000.96k0-0.28and the matricesA=4j-1.2-1.6j0.6j0.8i1.6i+1.2k2-0.8-2.4i1.2k-1.6-3.2-0.6i,B=-1.6+1.2j3.8j-1.6+1.2j-1.6i0.6-3.2i1.6+1.2k0.8j+2.4k-1.6-1.2k,C=-0.6-0.8i1.6j-0.4+1.2k-0.6+0.8j0.8j+0.6k1.31.6-0.3k-0.6i+0.8k,D=-0.8-0.2i1.6j+0.4k0.32-1.52i+0.24j012i-0.04j+0.64k0.48-0.4j-0.8+0.24k0.3i-0.64k

Acknowledgements

The authors thank a referee very much for his or her valuable suggestions and comments, which resulted in a great improvement of the original manuscript.

References (33)

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This research was supported by the grants from Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001), the Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (09YZ13), the Scientific Research Innovation Foundation of Shanghai University (A10010110003), and Shanghai Leading Academic Discipline Project (J50101).

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