Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications

https://doi.org/10.1016/j.amc.2006.06.012Get rights and content

Abstract

Assume that the linear quaternion matrix expression f(X1, X2) = A  A3X1B3  A4X2B4 where X1, X2 are variant quaternion matrices. In this paper, we derive the maximal and minimal ranks of f(X1, X2) subject to consistent systems of quaternion matrix equations A1X1 = C1, X1B1 = C2 and A2X2 = C3, X2B2 = C4. Moreover, corresponding results on some special cases are presented. As applications, we give necessary and sufficient conditions for the existence of solutions to some systems of quaternion matrix equations. Some previous known results can be regarded as the special cases of this paper.

Introduction

Extremal ranks of matrix expressions is one of important topics in matrix analysis and applications. 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. [7], [8], [9], [10], [11], [12]).

Let the linear matrix expressionf(X1,X2)=A-A3X1B3-A4X2B4,where A, A3, A4, B3, B4 are known matrices and X1, X2 are variant. In general, the matrix f(X1, X2) is variant for different choice of X1 and X2. Hence, the rank of f(X1, X2) may have different values with respect to the choice of X1 and X2. Because the rank of matrix is an integer between 0 and the minimum of row and column numbers of the matrix, extremal ranks, i.e., maximal and minimal ranks, of f(X1, X2) must exist with the variant matrices. Many problems in matrix theory and applications are closed related to extremal ranks of matrix expressions with variant entries. For example, a matrix equation A = A3X1 B3 + A4X2B4 is consistent if and only if the minimal rank of f(X1, X2) with respect to X1 and X2 are zero. In [6], [7], Tian has investigated the extremal ranks of f(X1, X2) over a field.

Uhlig in [1] presented the extremal ranks of solutions to the matrix equation AX = B. Mitra in [2] considered solutions with fixed ranks for the matrix equations AX = B and AXB = C. Tian in [3] has investigated the extremal rank solutions to the complex matrix equation AXB = C and gave some applications. Mitra [4] gave common solutions of minimal rank of the pair of complex matrix equations AX = C, XB = D, and studied the minimal ranks of common solutions to the pair of matrix equations A1XB1 = C1 and A2XB2 = C2 over a general field in [5].

The real quaternion matrices play a role in computer science, quantum physics, signal and color image processing, and so on (e.g. [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]).

Motivated by the work mentioned above and keeping some applications and interests of quaternion matrices in view, we in this paper investigate the extremal ranks of f(X1, X2) subject to consistent systems of linear matrix equations over HA1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4and its applications. In Section 2, we derive the formulas of extremal ranks of f(X1, X2) subject to the consistent systems (1.2), (1.3)). As applications, we in Section 3 give necessary and sufficient conditions for the existence of general solutions to the systems of linear matrix equations over H, the real quaternion field,A1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4,A3X1B3+A4X2B4=AandA1X=C1,XB1=C2,A3XB3=A,which has been studied by many authors (see, e.g. [13], [26], [27], [28], [29], [30], [31]). It is worthy to say that the system (1.4) can be used to investigate the centrosymmetric solution to the system (1.5), the symmetric solution to the systemA1X1=C1,X1A1=C2,A2X2=C3,X2A2=C4,A3X1A3+A4X2A4=A,over H which can be used to consider the bisymmetric solution to the systemAaX=Ca,XAa=Cb,AcXAc=A.We also give further research topics related to this paper in Section 4.

Throughout, we denote the set of all m × n matrices over the quaternion fieldH={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 rank of a matrix A over H by r(A), an inner inverse of a matrix A by A which satisfies AAA = A. Moreover, RA and LA stand for the two projectors LA = I  AA, RA = I  AA induced by A.

Section snippets

Main results

We begin with the following lemma which proof just like one over the complex field.

Lemma 2.1

Let A1Hm×n,B1Hr×s, C1Hm×r,C2Hn×s be known and X1Hn×r unknown. Then the following statements are equivalent:

  • (a)

    The system (1.2) is consistent.

  • (b)

    RA1C1=0,C2LB1=0,A1C2=C1B1.

  • (c)

    A1C2=C1B1,r[A1,C1]=r(A1),rB1C2=r(B1).

In that case, the general solution of (1.2) isX1=A1-C1+LA1C2B1-+LA1Y1RB1,where Y1 is an arbitrary matrix over H with appropriate dimension.

The following lemma is due to Marsglia and Styan [14], which can be

Solvable conditions to some systems of matrix equations

In this section, we use Theorem 2.5 to give necessary and sufficient conditions for the consistency to the systems (1.4), (1.5).

Theorem 3.1

Let AHl×u, A1Hm×n, B1Hr×s, C1Hm×r, C2Hn×s, A2Ht×p, B2Hq×k, C3Ht×q, C4Hp×k, A3Hl×n, B3Hr×u, A4Hl×p, B4Hq×u be known and X1Hn×r, X2Hp×q unknown. Then the system (1.4) is consistent if and only if the following equalities are all satisfied:A1C2=C1B1,A2C4=C3B2,rA1,C1=r(A1),rA2,C3=r(A2),rB1C2=r(B1),rB2C4=r(B2),rA10C1B3A3A4C4A0B2B4=rA100A3000B2B4,rA20C3B4A4A3C2

Conclusion

In this paper, we have derived the maximal and minimal ranks of the quaternion matrix expression A  A3X1B3  A4X2B4 subject to consistent matrix equations A1X1 = C1, X1B1 = C2 and A2X2 = C3, X2B2 = C4. Using the results, we give some solvable conditions to some systems of quaternion matrix equations. Some known results can be viewed as the special cases of this paper. Similarly, we can study the maximal and minimal ranks of A  A3X1B3  A4X2B4 subject to consistent matrix equations A1X1 = C1, A2X1 = C2 and X2B1 = C3

Acknowledgements

This research was supported by the Natural Science Foundation of China (No. 10471085), the Natural Science Foundation of Shanghai, the Development Foundation of Shanghai Educational Committee (No. 214498), and the Special Funds for Major Specialties of Shanghai Education Committee.

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