- Email: [email protected]
Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications
Available online at www.sciencedirect.com
Applied Mathematics and Computation 195 (2008) 733–744 www.elsevier.com/locate/amc
Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications Qing-Wen Wang
a,*
, Shao-Wen Yu a, Chun-Yan Lin
q
b
a
b
Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, PR China School of Statistics and Sciences, Shandong Finance University, 40 Shungeng Road, Jinan 250014, PR China
Abstract In this paper, we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression C 4 A4 XB4 where X is a variant quaternion matrix subject to quaternion matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 ; A3 XB3 ¼ C 3 . As applications, we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 ; A3 XB3 ¼ C 3 ; A4 XB4 ¼ C 4 , which was investigated by Wang [Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin., 21(2) (2005) 323–334], by rank equalities. In addition, extremal ranks of the generalized Schur complement D CA B with respect to an inner inverse A of A, which is a common solution to quaternion matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 , are also considered. Some previous known results can be viewed as special cases of the results of this paper. Ó 2007 Elsevier Inc. All rights reserved. Keywords: System of quaternion matrix equations; Minimal rank; Maximal rank; Linear matrix expression; Generalized inverse; Schur complement
1. Introduction Throughout this paper, we denote the real number field by R, the set of all m n matrices over the quaternion algebra H ¼ fa0 þ a1 i þ a2 j þ a3 kji2 ¼ j2 ¼ k 2 ¼ ijk ¼ 1; a0 ; a1 ; a2 ; a3 2 Rg by Hmn , the identity matrix with the appropriate size by I, the column right space, the row left space of a matrix A over H by RðAÞ, NðAÞ, respectively, the dimension of RðAÞ by dimRðAÞ, an inner inverse of a matrix A by A which satisfies AA A ¼ A, a reflexive inverse of matrix A over H by Aþ which satisfies simultaneously
q
This research was supported by the Natural Science Foundation of China (60672160), the Doctoral Program Foundation of Shanghai University, and the Special Funds for Major Specialities of Shanghai Education Committee. * Corresponding author. E-mail addresses: [email protected] (Q.-W. Wang), [email protected] (C.-Y. Lin). 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.018
734
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
AAþ A ¼ A and Aþ AAþ ¼ Aþ . Moreover, RA and LA stand for the two projectors LA ¼ I Aþ A, RA ¼ I AAþ 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 [1], for a quaternion matrix A, dimRðAÞ ¼ dim NðAÞ. dimRðAÞ is called the rank of a quaternion matrix A and denoted by r(A). Mitra in [2] investigated the system of matrix equations A1 X ¼ C 1 ;
XB2 ¼ C 2 :
ð1:1Þ
Bhimasankaram in [3] gave a necessary and sufficient condition for the system of matrix equations A1 X ¼ C 1 ;
XB2 ¼ C 2 ;
A3 XB3 ¼ C 3
ð1:2Þ
to have a general solution and presented the representation of the general solution. Lin and Wang in [4] gave a practical solvability condition and a new expression of system (1.2), and studied the extremal ranks of the general solution to (1.2) in [5]. Wang [6] established necessary and sufficient conditions for the existence and the expression of the general solution to the system of matrix equations by generalized inverses, A1 X ¼ C 1 ;
XB2 ¼ C 2 ;
A3 XB3 ¼ C 3 ;
A4 XB4 ¼ C 4
ð1:3Þ
and presented its applications. In view of the complicated computations of the generalized inverses of matrices, we naturally hope to establish a practical solvability condition of the system (1.3) by rank equalities. In 2003, Tian and Cheng in [7] investigated the maximal and minimal ranks of A BXC with respect to X with applications, and gave in [8] the maximal and minimal ranks of A1 B1 XC 1 subject to a consistent matrix equation B2 XC 2 ¼ A2 . We have an observation that by investigating extremal ranks of the matrix expression f ðX Þ ¼ C 4 A4 XB4
ð1:4Þ
subject to the consistent system of matrix equations (1.2), a new necessary and sufficient condition to system (1.3) can be given, which is more practical than one given by generalized inverses. It is well known that Schur complement is one of the most important matrix expressions in matrix theory, there have been many results in the literature on Schur complements and their applications (see, e.g. [9–12]). Tian in [8,13] has investigated the maximal and minimal ranks of Schur complements with applications. Motivated by the work mentioned above and keeping applications and interests of quaternion matrices in view (see, e.g. [14–28]), we in this paper investigate the extremal ranks of the quaternion matrix expression (1.4) subject to the consistent system of quaternion matrix equations (1.2) and its applications. In Section 2, we derive the formulas of extremal ranks of (1.4) subject to (1.2). As applications, in Section 3, we give a practical solvability condition to system (1.3) by rank equalities. Further using the results, we establish new necessary and sufficient conditions for solvability to the systems over H A1 X ¼ C 1 ;
A2 X ¼ C 2 ;
A3 XB3 ¼ C 3 ;
A4 XB4 ¼ C 4
ð1:5Þ
XB1 ¼ C 1 ;
XB2 ¼ C 2 ;
A3 XB3 ¼ C 3 ;
A4 XB4 ¼ C 4 ;
ð1:6Þ
and which were investigated in [25,31]. In Section 4, we derive extremal ranks of generalized Schur complement subject to (1.1). A necessary and sufficient condition for the invariance of the rank of (1.4) is given in Section 5. Some previous known results can be regarded as the special cases of this paper. 2. Extremal ranks of (1.4) subject to system (1.2) By Lemma 2.2 in [6] and Lemma 2.1 in [29], we can obtain the following. Lemma 2.1. Let A1 2 Hmn , B2 2 Hrs , C 1 2 Hmr , C 2 2 Hns be known and X 1 2 Hnr unknown. Then the following statements are equivalent: (1) The system (1.1) is consistent. (2) RA1 C 1 ¼ 0; C 2 LB2 ¼ 0; A1 C 2 ¼ C 1 B2 :
ð2:1Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
(3) A1 C 2 ¼ C 1 B2 ; r½A1 ; C 1 ¼ rðA1 Þ; r
B2 C2
¼ rðB2 Þ:
735
ð2:2Þ
Lemma 2.2 (See Theorem 2.4 in [4]). Suppose that A1 2 Hmn , A3 2 Hkn , B2 2 Hrs , B3 2 Hrp , C 1 2 Hmr , C 2 2 Hns , C 3 2 Hkp are known, X 2 Hnr unknown, and K ¼ A3 LA1 , H ¼ RB2 B3 , Q1 ¼ C 3 A3 Aþ 1 C 1 B3 þ þ KC 2 Bþ 2 B3 , K Q1 H ¼ Q2 , then the following statements are equivalent: (1) The system (1.2) is consistent. (2) Q1 ¼ KQ2 H ; þ þ þ A1 A þ 1 C 1 ¼ C 1 ; C 2 B 2 B2 ¼ C 2 ; A1 C 2 ¼ C 1 B 2 ; A3 A 3 C 3 B3 B3 ¼ C 3 : (3) The equalities in (2.2) hold and B3 r½ A3 C 3 ¼ rðA3 Þ; r ¼ rðB3 Þ; C3 A1 C 1 B 3 B3 A1 B2 r ¼r ; r ¼ r ½ B2 A3 C3 A3 A3 C 2 C 3
ð2:3Þ ð2:4Þ
ð2:5Þ B3 :
ð2:6Þ
In that case, the general solution of (1.2) can be expressed as þ X ¼ Aþ 1 C 1 þ LA1 C 2 B2 þ LA1 Q2 RB2 þ LA1 LK ZRB2 þ LA1 WRH RB2 ;
where Z; W are arbitrary matrices over H with compatible sizes. Lemma 2.3 (Lemma 2.4 in [27]). Let A 2 Hmn , B 2 Hmk , C 2 Hln , D 2 Hjk and E 2 Hli . Then they satisfy the following rank equalities A (a) rðCLA Þ ¼ r rðAÞ. C B A (b) r½ B ALC ¼ r rðCÞ. 0 C C C 0 (c) r ¼r rðBÞ. RB A A B 2 3 A B 0 A BLD (d) r ¼ r4 C 0 E 5 rðDÞ rðEÞ. RE C 0 0 D 0 Lemma 2.3 plays an important role in simplifying ranks of various block matrices. Tian in [8] has given the following Lemma over a field. The result can be generalized to H. Lemma 2.4. Let A 2 Hmn , B1 2 Hmp1 , B2 2 Hmp2 , C 1 2 Hq1 n and C 2 2 Hq2 n be given with RðB1 Þ RðB2 Þ and NðC 2 Þ NðC 1 Þ. Then A A B1 max rðA B1 X 1 C 1 B2 X 2 C 2 Þ ¼ min r½A; B2 ; r ;r ; ð2:7Þ X 1 ;X 2 C1 C2 0 A A B1 A B1 A B2 þr min rðA B1 X 1 C 1 B2 X 2 C 2 Þ ¼ r½A; B2 þ r r r : ð2:8Þ X 1 ;X 2 C1 C2 0 C1 0 C2 0 Now we consider the extremal ranks of the matrix expression (1.4) subject to the consistent system (1.2). Theorem 2.5. The extremal ranks of the quaternion matrix expression f ðX Þ defined as (1.4) subject to the consistent system (1.2) are the following: max r½f ðX Þ ¼ minfa; b; cg;
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
ð2:9Þ
736
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
where
"
a¼r
A1
C4
A4
B2
B4
A4 C 2
C4
A1
0
0
A3 C 2
C 3
0
0
B2
B3
" b¼r 2
#
C 1 B4
6 6 A3 c ¼ r6 6A 4 4 0
rðA1 Þ; # rðB2 Þ;
" min r½f ðX Þ ¼ r
C 1 B4
" # 7 A 0 7 7 r 1 r ½ B2 7 C4 5 A3 B4
C 1 B4
A1
C4
A4
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
3
2
A1 6 6 A3 r6 6A 4 4 0
B3 ; 2
#
" þr
0 A3 C 2 0 B2
B2
B4
A4 C 2
C4
C 1 B4
#
0
0
A3 C 2
C 3
0
0
B2
B3
A1
6 6 A3 þ r6 6A 4 4 0
3
2 A1 7 0 7 6 7 r 4 A4 C4 7 5 0 B4
3
7 0 7 7 C4 7 5 B4
3
0
0
0
0
7 C 4 5:
B2
B3
B4
C 1 B4
C 1 B4
ð2:10Þ
Proof. By Lemma 2.2, the general solution of the system (1.2) can be expressed as þ X ¼ Aþ 1 C 1 þ LA1 C 2 B2 þ LA1 Q2 RB2 þ LA1 LK ZRB2 þ LA1 WRH RB2 ;
ð2:11Þ
where W ; Z are arbitrary matrices over H with appropriate sizes. Substituting (2.11) into (1.4) yields þ f ðX Þ ¼ C 4 A4 Aþ 1 C 1 B4 A4 LA1 C 2 B2 B4 A4 LA1 Q2 RB2 B4 A4 LA1 LK ZRB2 B4 A4 LA1 WRH RB2 B4 :
Put þ C 4 A4 Aþ 1 C 1 B4 A4 LA1 C 2 B2 B4 A4 LA1 Q2 RB2 B4 ¼ C;
ð2:12Þ
A4 LA1 LK ¼ D1 ;
ð2:13Þ
A4 LA1 ¼ D2 ;
RB2 B4 ¼ E1 ;
RH RB2 B4 ¼ E2 ;
then f ðX Þ ¼ C D1 ZE1 D2 WE2 :
ð2:14Þ
Note that RðD1 Þ RðD2 Þ, NðE2 Þ NðE1 Þ. Thus, applying (2.7) and (2.8) to (2.14), we get the following: " # " ( #) C C D1 max r½f ðX Þ ¼ max rðC D1 ZE1 D2 WE2 Þ ¼ min r½ C D2 ; r ;r ; ð2:15Þ Z;W A1 X ¼C 1 E1 E2 0 XB ¼C 2
2
A3 XB3 ¼C 3
min r½f ðX Þ ¼ min rðC D1 ZE1 D2 WE2 Þ
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
Z;W
¼ r½ C
C C D2 þ r þr E1 E2
D1 C r 0 E1
Now we simplify the ranks of block matrices in (2.15) and (2.16).
D1 C r 0 E2
D2 : 0
ð2:16Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
737
In view of Lemma 2.3, block Gaussian elimination, (2.3), (2.4), (2.12) and (2.13), we have the following: " # A1 rðKÞ ¼ rðA3 LA1 Þ ¼ r rðA1 Þ; A3 rðH Þ ¼ rðRB2 B3 Þ ¼ r½ B2 B3 rðB2 Þ; C A4 LA1 LK C D1 r ¼r RH RB2 B4 0 E2 0 2 3 C A4 LA1 0 6 7 0 H 5 rðKÞ rðH Þ ¼ r4 RB2 B4 0 K 0 2 3 C A4 LA1 0 6 7 0 RB2 B3 5 rðKÞ rðH Þ ¼ r4 RB2 B4 0 A3 LA1 0 2 3 C A4 0 0 6B 7 6 4 0 B3 B2 7 ¼ r6 7 rðKÞ rðH Þ rðA1 Þ rðB2 Þ 4 0 A3 0 0 5 2
0 A1 0 C4 A4
6 B 6 4 ¼ r6 4 0
0 A3
0 0 B3 C 3
C 1 B4 A1 0 A1 0 0 6 A A C C 3 2 3 6 3 ¼ r6 4 A4 0 0 2
0
3
B2 7 7 7 rðKÞ rðH Þ rðA1 Þ rðB2 Þ A3 C 2 5 0 3 C 1 B4 0 7 A1 7 r ½ B2 7r C4 5 A3
B3 :
ð2:17Þ
0 B2 B3 B4 Similarly, we can obtain the following: C 1 B4 A1 r½ C D2 ¼ r rðA1 Þ; C4 A4 C B2 B4 r ¼r rðB2 Þ; E1 A4 C 2 C 4 2 3 0 C 1 B4 A1 6 A A C C D1 A1 0 7 3 2 6 3 7 r r rðB2 Þ; ¼ r6 7 4 A4 E1 0 0 C4 5 A3
ð2:19Þ
0 B2 B4 3 A1 0 0 C 1 B4 C D2 6 7 C 4 5 r½ B2 B3 rðA1 Þ: r ¼ r 4 A4 0 0 E2 0 0 B2 B 3 B4 Substituting (2.17)–(2.21) into (2.15) and (2.16) yields (2.9) and (2.10). h
ð2:21Þ
ð2:18Þ
ð2:20Þ
2
In Theorem 2.5, letting C4 vanish and A4 and B4 be I and I with appropriate sizes, respectively, we have the following. Corollary 2.6. Assume that A1 2 Hmn , A3 2 Hkn , B2 2 Hrs , B3 2 Hrp , C 1 2 Hmr , C 2 2 Hns , C 3 2 Hkp are given, then the maximal and minimal ranks of the general solution X of the consistent system (1.2) can be expressed as
738
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
max rðX Þ ¼ minfa; b; cg where a ¼ n þ rðC 1 Þ rðA1 Þ; b ¼ r þ rðC 2 Þ rðB2 Þ; A1 C 2 C 1 B3 A1 c¼r r r½ B2 B3 þ n þ r; A3 C 2 C3 A3 A 1 C 2 C 1 B3 A1 C 2 min rðX Þ ¼ r r r½ C 1 B2 C 1 B3 þ rðC 1 Þ þ rðC 2 Þ: A3 C 2 C3 A3 C 2 Remark 2.1. Corollary 2.6 is the main results of [5]. In Theorem 2.5, assuming A1 ; B2 ; C 1 and C2 vanish, we have the following. Corollary 2.7. Suppose that the matrix equation A3 XB3 ¼ C 3 is consistent, then the extremal ranks of the quaternion matrix expression f(X) defined as (1.4) subject to A3 XB3 ¼ C 3 are the following: 9 8 2 3 A3 C 3 0 > > = < B4 6 7 0 C 4 5 rðA3 Þ rðB3 Þ ; max r½f ðX Þ ¼ min r½ C 4 A4 ; r ; r 4 A4 A3 XB3 ¼C 3 > > C4 ; : 0 B3 B4 2 3 2 3 A3 0 A3 C 3 0 A4 0 C 4 B4 6 7 6 7 min r½f ðX Þ ¼ r½ C 4 A4 þ r 0 C 4 5 r 4 A4 C 4 5 r þ r 4 A4 : A3 XB3 ¼C 3 C4 0 B 3 B4 0 B3 B4 0 B4 Remark 2.2. Corollary 2.7 is Theorem 3.2 in [8]. 3. A practical solvability condition of system (1.3) In this section, we use Theorem 2.5 to give a necessary and sufficient condition for the consistency to the system (1.3) by rank equalities. Theorem 3.1. Let A1 2 Hmn , C 1 2 Hmr , B2 2 Hrs , C 2 2 Hns , A3 2 Hkn , B3 2 Hrp , C 3 2 Hkp , A4 2 Hqn , B4 2 Hrl , C 4 2 Hql be known and X 2 Hnr unknown, then the system (1.3) is consistent if and only if (2.2), (2.5) and (2.6) hold, and the following equalities are all satisfied: B4 r½ A4 C 4 ¼ rðA4 Þ; r ð3:1Þ ¼ rðB4 Þ; C4 B4 B2 ð3:2Þ r ¼ r½ B2 B4 ; A4 C 2 C 4 C 1 B4 A1 A1 r ¼r ; ð3:3Þ C4 A4 A4 2 3 2 3 0 0 C 1 B4 A1 A1 6 A A C C 7 0 7 3 2 3 6 3 6 7 ð3:4Þ r6 7 ¼ r4 A3 5 þ r½ B2 B3 B4 : 4 A4 0 0 C4 5 A4 0 B2 B3 B4 Proof. It is obvious that the system (1.3) is consistent if and only if the system (1.2) is consistent and min r½f ðX Þ ¼ 0;
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
ð3:5Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
739
where f(X) is defined as (1.4). Let X0 be a solution to the system (1.3). Then X0 is a solution to system (1.2) and X0 satisfies A4 X 0 B4 ¼ C 4 . Hence, Lemma 2.2 yields (2.2), (2.5), (2.6) and (3.1). It follows from 2
0 0
I
6 60 6 60 4 0
I
0
0
I
0
32
A1
76 A3 X 0 7 6 A3 76 6 0 7 5 4 A4 I 0
0 0
0
0
A3 C 2
C 3
0
0
B2
B3
C 1 B4
32
I
76 0 76 0 76 6 C4 7 54 0 0 B4
0
0 X 0 B4
I
0
0
0
I
0
0
0
I
3
2
A1
7 6 7 6 A3 7¼6 7 6A 5 4 4 0
0
0
0
0
0
0
B2
B3
0
3
7 0 7 7 0 7 5 B4
that (3.4) holds. Similarly, we can obtain (3.2) and (3.3). Conversely, assume that (2.2), (2.5) and (2.6) hold, then by Lemma 2.2, system (1.2) is consistent. By (2.10), (3.2)–(3.4), and noting that 2
0
A1
6A 6 3 r6 4 A4
A3 C 2 0
0 A1 6 r 4 A4
B2 0 0
0 0
0
B2
B3
2
3
2 3 A1 7 0 7 6 7 7 P r4 A3 5 þ r½ B2 B4 ; C4 5 A4 B4 3 C 1 B4 A1 7 C4 5 P r þ r ½ B2 B3 B4 A4 B4
C 1 B4
yields min r½f ðX Þ 6 0:
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
However, min r½f ðX Þ P 0:
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
Hence (3.5) holds, implying the system (1.3) is consistent.
h
In Theorem 3.1 and its proof, let B2 ; C 2 vanish and A1 C1 A1 ¼ ; C1 ¼ : A2 C2 Then we can get the following. Corollary 3.2. The system of quaternion matrix equations (1.5) is consistent if and only if (2.5), (3.1) hold and the following equalities are all satisfied: " # " # A1 C 1 A1 r ¼r ; A2 C 2 A2 2 3 2 3 A1 A1 C 1 B 3 6 7 6 7 r 4 A2 C 2 B 3 5 ¼ r 4 A2 5 ; 2
A3 C 1 B4
6 r 4 C 2 B4 C4
C3 A1
3
2
A3 A1
3
7 6 7 A 2 5 ¼ r 4 A2 5 ; A4
A4
740
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
2
A1
6 6 A2 6 r6 6 A3 6 4 A4 0
0 0 C 3 0 B3
C 1 B4
3
2 3 A1 7 C 2 B4 7 6A 7 7 6 27 0 7 7 þ r ½ B3 7 ¼ r6 4 A3 5 7 C4 5 A4 B4
B4 :
Remark 3.1. Corollary 3.2 gives a practical solvability condition to system (2.5) which was investigated in [25]. In Theorem 3.1 and its proof, letting A1 ; C 1 vanish and putting B2 ¼ ½ B1 following.
B2 and C 2 ¼ ½ C 1
C 2 yields the
Corollary 3.3. The system of quaternion matrix equations (1.6) has a solution if and only if (2.5), (3.1) hold and the following equalities are all satisfied: B1 B 2 r ¼ r½ B1 B2 ; C1 C2 B1 B2 B3 r ¼ r½ B1 B2 B3 ; A3 C 1 A3 C 2 C 3 B1 B2 B4 r ¼ r½ B1 B2 B4 ; A4 C 1 A4 C 2 C 4 2 3 A3 A3 C 1 A3 C 2 C 3 0 A3 6 7 r 4 A4 0 0 0 C4 5 ¼ r þ r½ B1 B2 B3 B4 : A4 0 B1 B2 B3 B4 Remark 3.2. Corollary 3.3 presents a practical necessary and sufficient condition for the solvability to system (1.6) which was studied in [31]. Corollary 3.4 ([29,30]). Suppose that A3 ; B3 ; C 3 ; A4 ; B4 and C4 are those in Theorem 3.1, then the system of quaternion matrix equations A3 XB3 ¼ C 3 , A4 XB4 ¼ C 4 is consistent if and only if (2.5), (3.1) hold and the following equality is satisfied: 2 3 2 3 A3 C 3 0 A3 0 0 6 7 6 7 r 4 A4 0 C 4 5 ¼ r4 A4 0 0 5: 0
B3
B4
0
B3
B4
By Theorem 3.1, we can also get the following. Corollary 3.5. Let A1 2 Hmn , C 1 2 Hmr , B2 2 Hrs , C 2 2 Hns and A; B 2 Hrn . Then A and B have a common inner inverse which is a solution of the system (1.1) if and only if (2.2) holds and the following equalities are all satisfied: B2 A r ¼ r½ B2 A ; AC 2 A B2 B r ¼ r½ B2 B ; BC 2 B A1 C 1 A A1 r ; ¼r A A A C 1 B A1 A1 r ¼r ; B B B
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
2
A1
6A 6 r6 4B 0
0
0
AC 2 0
A 0
B2
A
3
2 3 A1 7 0 7 6 7 ¼ r 7 4 A 5 þ r ½ B2 B 5 B B
C1B
741
A
B :
4. Extremal ranks of schur complement subject to (1.1) As is well known, for a given block matrix A B M¼ ; C D where A; B; C and D are quaternion matrices with appropriate sizes, a generalized Schur complement of A in M is defined as S A ¼ D CA B;
ð4:1Þ
where A is an inner inverse of A, i.e. A 2 fX j AXA ¼ Ag. Now we use Theorem 2.5 to establish the extremal ranks of SA given by (4.1) with respect to A which is a solution to system (1.1). Theorem 4.1. Suppose A1 2 Hmn , C 1 2 Hmr , B2 2 Hrs , C 2 2 Hns , D 2 Hql , C 2 Hqn , B 2 Hrl , A 2 Hrn are given and system (1.1) is consistent, then the extreme ranks of SA given by (4.1) with respect to A which is a solution of (1.1) are the following: max rðS A Þ ¼ minfa; b; cg;
A1 A ¼C 1 A B2 ¼C 2
where a¼r
C1B
D B2
A1
C B
rðA1 Þ;
rðB2 Þ; CC 2 D 2 3 A1 0 0 C1B 6 A AC A A1 0 7 2 6 7 r c ¼ r6 r½ B2 A ; 7 4C 0 0 D 5 A 0 B2 A B 2 0 0 A1 6 A AC A C 1 B A1 B2 B 2 6 min rðS A Þ ¼ r þr þ r6 A1 A ¼C 1 4 CC C 0 0 D C D 2 A B2 ¼C 2 0 B2 A 2 3 2 3 A1 0 C1B A1 0 0 C 1 B 6 A AC 0 7 2 6 7 6 7 r6 D 5: 7 r4 C 0 0 4C 0 D 5 0 B2 A B 0 B2 B b¼r
Proof. It is obvious that max rðD CA BÞ ¼ max rðD CXBÞ;
A1 A ¼C 1 A B2 ¼C 2
A1 X ¼C 1 XB2 ¼C 2 AXA¼A
C1B
3
0 7 7 7 D 5 B
742
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
min rðD CA BÞ ¼ min rðD CXBÞ:
A1 A ¼C 1 A B2 ¼C 2
A1 X ¼C 1 XB2 ¼C 2 AXA¼A
Thus in Theorem 2.5 and its proof, letting A3 ¼ B3 ¼ C 3 ¼ A, A4 ¼ C, B4 ¼ B, C 4 ¼ D, we can easily get the proof. h In Theorem 4.1, let A1 ; C 1 ; B2 ; C 2 vanish. Then we can easily get the following. Corollary 4.2. The extreme ranks of SA given by (4.1) with respect to A are the following: ( " # " # ) B A B max rðS A Þ ¼ min r½ C D ; r ;r rðAÞ ; A D C D 2 3 A 0 " # " # " # B A B A 0 B 6 7 7 min rðS Þ ¼ rðAÞ þ r½ C D þ r þ r r r6 A 4 0 B 5: A D C D 0 C D C D Remark 4.1. Corollary 4.2 is Theorem 4.1 in [8]. 5. The rank invariance of (1.4) As another application of Theorem 2.5, we in this section consider the rank invariance of the matrix expression (1.4) subject to the consistent system (1.2). Theorem 5.1. Suppose that (1.2) is consistent, then the rank of f(X) defined by (1.4) subject to (1.2) is invariant if and only if 2
0
A1
6 6 A3 6 r6 6 A4 4
A3 C 2 0
0 2
or
B2 0
C 1 B4
0
0
C4
0
B2
B3
A1 6 r 4 A4
0 0
0 0
0 A1 6A 6 3 r6 4 A4
B2
0
B3 0 A3 C 2 0 B2
B4
#
" þr
C4
A1
# ;
A3
B4
0
2
3
7 " B2 0 7 7 7¼r C4 7 A4 C 2 5
A1 6 r6 4 A4
2
or
C 1 B4
B4
3
2
A1
6 " # 6 A3 A1 7 6 7þr ¼ r6 5 6 A4 A3 4 0
0
0
A3 C 2
C 3
0
0
B2
B3
C 1 B4
3 C 1 B4 C 1 B4 A1 7 C4 5 ¼ r þ r½ B2 B3 ; C4 A4 B4 3 2 A1 C 1 B4 0 0 7 6 0 7 6 A3 A3 C 2 C 3 7 þ r ½ B2 B3 ¼ r 6 4 A4 C4 5 0 0 B4 0 B2 B3
3
7 0 7 7 7 þ rðA1 Þ C4 7 5
ð5:1Þ
B4
3 C 1 B4 0 7 7 7 þ rðB2 Þ C4 5 B4
ð5:2Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
2
0
A1
6 6 A3 r6 6A 4 4
A3 C 2 0
0 2
C 1 B4
B2
3
" 7 B2 0 7 7¼r 7 C4 5 A4 C 2 B4
A1
0
0
C 1 B4
6 r 4 A4
0
0
C4
0
B2
B3
B4
743
B4
# þr
C4
3
" C 1 B4 7 5¼r C4
A1
"
A1 A3
# ;
#
A4
þ r ½ B2
B3 :
ð5:3Þ
Proof. It is obvious that the rank of f(X) subject to the consistent system (1.2) is invariant if and only if max r½f ðX Þ min r½f ðX Þ ¼ 0:
ð5:4Þ
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
By (5.4), Theorem 2.5 and simplifications, we can get (5.1), (5.2) and (5.3). h Corollary 5.2. The rank of SA defined by (4.1) with respect to A which is a solution to system (1.1) is invariant if and only if 2
6A 6 r6 4C 0 2
or
0
C1B
AC 2 0
0 D
B2
B
A1
A1 6 r4 C
0 0
0 0
0
B2
A
A1
0
0
2
6 r4 C 0 2 A1 6A 6 r6 4C
3 7 B2 7 7¼r 5 CC 2
B A1 þr ; D A
2 3 A1 C1B 6 A1 6A 7 D 5þr ¼ r6 4C A B 0
C1B
0 AC 2
0 A
0
0
B2
A
B2
B
3
A1
0
C1B
6A 6 r6 4C
AC 2 0
0 D
0 A1
B2 0 0
0
B2
or 2
2
6 r4 C 0
B
C 1 B A1 7 0 0 D 5¼r þ r½ B2 A ; D C B2 A B 3 2 A1 0 C1B 0 0 7 6 AC 2 0 7 6 A AC 2 A 7 þ r½ B2 A ¼ r6 4C 0 D 5 0 0
0
0 B2
0 A
3 C1B 0 7 7 7 þ rðA1 Þ D 5
3 7 B2 7 7¼r 5 CC 2
B 3 C1B C1B 7 D 5¼r D B
B A1 þr ; D A A1 þ r½ B2 C
A :
A
C1B 0 D B
3 7 7 7 þ rðB2 Þ 5
744
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
6. Conclusion In this paper, we have derived the maximal and minimal ranks of the quaternion matrix expression C 4 A4 XB4 subject to the consistent system of quaternion matrix equations A1 X 1 ¼ C 1 , X 1 B2 ¼ C 2 , A3 XB3 ¼ C 3 . Using the results, we give new and practical solvability conditions to systems of quaternion matrix Eqs. (1.3), (1.5) and (1.6). Moreover, maximal and minimal ranks of the generalized Schur complement D CA B with respect to an inner inverse A of A, which is a common solution to matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 , are also investigated. Some known results can be viewed as the special cases of this paper. References [1] T.W. Hungerford, Algebra, Spring-Verlag, New York Inc., 1980. [2] S.K. Mitra, The matrix equations AX ¼ C, XB ¼ D, Linear Algebra Appl. 59 (1984) 171–181. [3] P. Bhimasankaram, Common solutions to the linear matrix equations AX ¼ B; XC ¼ D, and EXF ¼ G, Sankhya Ser. A 38 (1976) 404–409. [4] C.Y. Lin, Q.W. Wang, New solvability conditions and a new expression of the general solution to a system of linear matrix equations over an arbitrary division ring, Southeast Asian Bull. Math. 29 (5) (2005) 755–762. [5] C.Y. Lin, Q.W. Wang, The minimal and maximal ranks of the general solution to a system of matrix equations over an arbitrary division ring, Math. Sci. Res. J. 10 (3) (2006) 57–65. [6] Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin. 21 (2) (2005) 323–334. [7] Y. Tian, S. Cheng, The maximal and minimal ranks of A BXC with applications, New York J. Math. 9 (2003) 345–362. [8] Y. Tian, Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear Algebra Appl. 355 (2002) 187– 214. [9] T. Ando, Generalized Schur complements, Linear Algebra Appl. 27 (1979) 173–186. [10] M. Fiedler, Remarks on the Schur complement, Linear Algebra Appl. 39 (1981) 189–195. [11] D. Carlson, E. Haynsworth, T. Markham, A generalization of the Schur complements by means of the Moore-Penrose inverse, SIAM J. Appl. Math. 26 (1974) 169–179. [12] D. Carlson, What are Schur complements, anyway? Linear Algebra Appl. 74 (1986) 257–275. [13] Y. Tian, More on maximal and minimal ranks of Schur complements with applications, Appl. Math. Comput. 152 (3) (2004) 675–692. [14] S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, Oxford, 1995. [15] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997) 21–57. [16] A. Baker, Right eigenvalues for quaternionic matrices: a topological approach, Linear Algebra Appl. 286 (1999) 303–309. [17] D.I. Merino, V. Sergeichuk, Littlewood’s algorithm and quaternion matrices, Linear Algebra Appl. 298 (1999) 193–208. [18] S. De Leo, G. Scolarici, Right eigenvalue equation in quaternionic quantum mechanics, J. Phys. A 33 (2000) 2971–2995. [19] D.R. Farenick, B.A.F. Pidkowich, The spectral theorem in quaternions, Linear Algebra Appl. 371 (2003) 75–102. [20] C.E. Moxey, S.J. Sangwine, T.A. Ell, Hypercomplex correlation techniques for vector images, IEEE Trans. Signal Process. 51 (7) (2003) 1941–1953. [21] N. Le Bihan, J. Mars, Singular value decomposition of matrices of quaternions: a new tool for vector-sensor signal processing, Signal Process. 84 (7) (2004) 1177–1199. [22] N. Le Bihan, S.J. Sangwine, Quaternion principal component analysis of color images, in: IEEE International Conference on Image Processing (ICIP), Barcelona, Spain, September, 2003. [23] N. Le Bihan, S.J. Sangwine, Color image decomposition using quaternion singular value decomposition, in: IEEE International Conference on Visual Information Engineering (VIE), Guildford, UK, July, 2003. [24] S.J. Sangwine, N. Le Bihan, Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations, Appl. Math. Comput. 182 (1) (2006) 727–738. [25] Q.W. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 665–675. [26] Q.W. Wang, Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 641–650. [27] Q.W. Wang, G.J. Song, C.Y. Lin, Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application, Appl. Math.Comput. 189 (2007) 1517–1532. [28] Q.W. Wang, Z.C. Wu, C.Y. Lin, Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Appl. Math. Comput. 182 (2006) 1755–1764. [29] Q.W. Wang, A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra Appl. 384 (2004) 43–54. [30] S.K. Mitra, A pair of simultaneous linear matrix equations A1 XB1 ¼ C 1 and A2 XB2 ¼ C 2 and a programming problem, Linear Algebra Appl. 131 (1990) 107–123. [31] Q.W. Wang, F. Qin, C.Y. Lin, The common solution to matrix equations over a regular ring with applications, Indian J. Pure Appl. Math. 36 (12) (2005) 655–672.
Applied Mathematics and Computation 195 (2008) 733–744 www.elsevier.com/locate/amc
Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications Qing-Wen Wang
a,*
, Shao-Wen Yu a, Chun-Yan Lin
q
b
a
b
Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, PR China School of Statistics and Sciences, Shandong Finance University, 40 Shungeng Road, Jinan 250014, PR China
Abstract In this paper, we establish the formulas of the maximal and minimal ranks of the quaternion matrix expression C 4 A4 XB4 where X is a variant quaternion matrix subject to quaternion matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 ; A3 XB3 ¼ C 3 . As applications, we give a new necessary and sufficient condition for the existence of solutions to the system of matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 ; A3 XB3 ¼ C 3 ; A4 XB4 ¼ C 4 , which was investigated by Wang [Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin., 21(2) (2005) 323–334], by rank equalities. In addition, extremal ranks of the generalized Schur complement D CA B with respect to an inner inverse A of A, which is a common solution to quaternion matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 , are also considered. Some previous known results can be viewed as special cases of the results of this paper. Ó 2007 Elsevier Inc. All rights reserved. Keywords: System of quaternion matrix equations; Minimal rank; Maximal rank; Linear matrix expression; Generalized inverse; Schur complement
1. Introduction Throughout this paper, we denote the real number field by R, the set of all m n matrices over the quaternion algebra H ¼ fa0 þ a1 i þ a2 j þ a3 kji2 ¼ j2 ¼ k 2 ¼ ijk ¼ 1; a0 ; a1 ; a2 ; a3 2 Rg by Hmn , the identity matrix with the appropriate size by I, the column right space, the row left space of a matrix A over H by RðAÞ, NðAÞ, respectively, the dimension of RðAÞ by dimRðAÞ, an inner inverse of a matrix A by A which satisfies AA A ¼ A, a reflexive inverse of matrix A over H by Aþ which satisfies simultaneously
q
This research was supported by the Natural Science Foundation of China (60672160), the Doctoral Program Foundation of Shanghai University, and the Special Funds for Major Specialities of Shanghai Education Committee. * Corresponding author. E-mail addresses: [email protected] (Q.-W. Wang), [email protected] (C.-Y. Lin). 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.018
734
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
AAþ A ¼ A and Aþ AAþ ¼ Aþ . Moreover, RA and LA stand for the two projectors LA ¼ I Aþ A, RA ¼ I AAþ 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 [1], for a quaternion matrix A, dimRðAÞ ¼ dim NðAÞ. dimRðAÞ is called the rank of a quaternion matrix A and denoted by r(A). Mitra in [2] investigated the system of matrix equations A1 X ¼ C 1 ;
XB2 ¼ C 2 :
ð1:1Þ
Bhimasankaram in [3] gave a necessary and sufficient condition for the system of matrix equations A1 X ¼ C 1 ;
XB2 ¼ C 2 ;
A3 XB3 ¼ C 3
ð1:2Þ
to have a general solution and presented the representation of the general solution. Lin and Wang in [4] gave a practical solvability condition and a new expression of system (1.2), and studied the extremal ranks of the general solution to (1.2) in [5]. Wang [6] established necessary and sufficient conditions for the existence and the expression of the general solution to the system of matrix equations by generalized inverses, A1 X ¼ C 1 ;
XB2 ¼ C 2 ;
A3 XB3 ¼ C 3 ;
A4 XB4 ¼ C 4
ð1:3Þ
and presented its applications. In view of the complicated computations of the generalized inverses of matrices, we naturally hope to establish a practical solvability condition of the system (1.3) by rank equalities. In 2003, Tian and Cheng in [7] investigated the maximal and minimal ranks of A BXC with respect to X with applications, and gave in [8] the maximal and minimal ranks of A1 B1 XC 1 subject to a consistent matrix equation B2 XC 2 ¼ A2 . We have an observation that by investigating extremal ranks of the matrix expression f ðX Þ ¼ C 4 A4 XB4
ð1:4Þ
subject to the consistent system of matrix equations (1.2), a new necessary and sufficient condition to system (1.3) can be given, which is more practical than one given by generalized inverses. It is well known that Schur complement is one of the most important matrix expressions in matrix theory, there have been many results in the literature on Schur complements and their applications (see, e.g. [9–12]). Tian in [8,13] has investigated the maximal and minimal ranks of Schur complements with applications. Motivated by the work mentioned above and keeping applications and interests of quaternion matrices in view (see, e.g. [14–28]), we in this paper investigate the extremal ranks of the quaternion matrix expression (1.4) subject to the consistent system of quaternion matrix equations (1.2) and its applications. In Section 2, we derive the formulas of extremal ranks of (1.4) subject to (1.2). As applications, in Section 3, we give a practical solvability condition to system (1.3) by rank equalities. Further using the results, we establish new necessary and sufficient conditions for solvability to the systems over H A1 X ¼ C 1 ;
A2 X ¼ C 2 ;
A3 XB3 ¼ C 3 ;
A4 XB4 ¼ C 4
ð1:5Þ
XB1 ¼ C 1 ;
XB2 ¼ C 2 ;
A3 XB3 ¼ C 3 ;
A4 XB4 ¼ C 4 ;
ð1:6Þ
and which were investigated in [25,31]. In Section 4, we derive extremal ranks of generalized Schur complement subject to (1.1). A necessary and sufficient condition for the invariance of the rank of (1.4) is given in Section 5. Some previous known results can be regarded as the special cases of this paper. 2. Extremal ranks of (1.4) subject to system (1.2) By Lemma 2.2 in [6] and Lemma 2.1 in [29], we can obtain the following. Lemma 2.1. Let A1 2 Hmn , B2 2 Hrs , C 1 2 Hmr , C 2 2 Hns be known and X 1 2 Hnr unknown. Then the following statements are equivalent: (1) The system (1.1) is consistent. (2) RA1 C 1 ¼ 0; C 2 LB2 ¼ 0; A1 C 2 ¼ C 1 B2 :
ð2:1Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
(3) A1 C 2 ¼ C 1 B2 ; r½A1 ; C 1 ¼ rðA1 Þ; r
B2 C2
¼ rðB2 Þ:
735
ð2:2Þ
Lemma 2.2 (See Theorem 2.4 in [4]). Suppose that A1 2 Hmn , A3 2 Hkn , B2 2 Hrs , B3 2 Hrp , C 1 2 Hmr , C 2 2 Hns , C 3 2 Hkp are known, X 2 Hnr unknown, and K ¼ A3 LA1 , H ¼ RB2 B3 , Q1 ¼ C 3 A3 Aþ 1 C 1 B3 þ þ KC 2 Bþ 2 B3 , K Q1 H ¼ Q2 , then the following statements are equivalent: (1) The system (1.2) is consistent. (2) Q1 ¼ KQ2 H ; þ þ þ A1 A þ 1 C 1 ¼ C 1 ; C 2 B 2 B2 ¼ C 2 ; A1 C 2 ¼ C 1 B 2 ; A3 A 3 C 3 B3 B3 ¼ C 3 : (3) The equalities in (2.2) hold and B3 r½ A3 C 3 ¼ rðA3 Þ; r ¼ rðB3 Þ; C3 A1 C 1 B 3 B3 A1 B2 r ¼r ; r ¼ r ½ B2 A3 C3 A3 A3 C 2 C 3
ð2:3Þ ð2:4Þ
ð2:5Þ B3 :
ð2:6Þ
In that case, the general solution of (1.2) can be expressed as þ X ¼ Aþ 1 C 1 þ LA1 C 2 B2 þ LA1 Q2 RB2 þ LA1 LK ZRB2 þ LA1 WRH RB2 ;
where Z; W are arbitrary matrices over H with compatible sizes. Lemma 2.3 (Lemma 2.4 in [27]). Let A 2 Hmn , B 2 Hmk , C 2 Hln , D 2 Hjk and E 2 Hli . Then they satisfy the following rank equalities A (a) rðCLA Þ ¼ r rðAÞ. C B A (b) r½ B ALC ¼ r rðCÞ. 0 C C C 0 (c) r ¼r rðBÞ. RB A A B 2 3 A B 0 A BLD (d) r ¼ r4 C 0 E 5 rðDÞ rðEÞ. RE C 0 0 D 0 Lemma 2.3 plays an important role in simplifying ranks of various block matrices. Tian in [8] has given the following Lemma over a field. The result can be generalized to H. Lemma 2.4. Let A 2 Hmn , B1 2 Hmp1 , B2 2 Hmp2 , C 1 2 Hq1 n and C 2 2 Hq2 n be given with RðB1 Þ RðB2 Þ and NðC 2 Þ NðC 1 Þ. Then A A B1 max rðA B1 X 1 C 1 B2 X 2 C 2 Þ ¼ min r½A; B2 ; r ;r ; ð2:7Þ X 1 ;X 2 C1 C2 0 A A B1 A B1 A B2 þr min rðA B1 X 1 C 1 B2 X 2 C 2 Þ ¼ r½A; B2 þ r r r : ð2:8Þ X 1 ;X 2 C1 C2 0 C1 0 C2 0 Now we consider the extremal ranks of the matrix expression (1.4) subject to the consistent system (1.2). Theorem 2.5. The extremal ranks of the quaternion matrix expression f ðX Þ defined as (1.4) subject to the consistent system (1.2) are the following: max r½f ðX Þ ¼ minfa; b; cg;
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
ð2:9Þ
736
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
where
"
a¼r
A1
C4
A4
B2
B4
A4 C 2
C4
A1
0
0
A3 C 2
C 3
0
0
B2
B3
" b¼r 2
#
C 1 B4
6 6 A3 c ¼ r6 6A 4 4 0
rðA1 Þ; # rðB2 Þ;
" min r½f ðX Þ ¼ r
C 1 B4
" # 7 A 0 7 7 r 1 r ½ B2 7 C4 5 A3 B4
C 1 B4
A1
C4
A4
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
3
2
A1 6 6 A3 r6 6A 4 4 0
B3 ; 2
#
" þr
0 A3 C 2 0 B2
B2
B4
A4 C 2
C4
C 1 B4
#
0
0
A3 C 2
C 3
0
0
B2
B3
A1
6 6 A3 þ r6 6A 4 4 0
3
2 A1 7 0 7 6 7 r 4 A4 C4 7 5 0 B4
3
7 0 7 7 C4 7 5 B4
3
0
0
0
0
7 C 4 5:
B2
B3
B4
C 1 B4
C 1 B4
ð2:10Þ
Proof. By Lemma 2.2, the general solution of the system (1.2) can be expressed as þ X ¼ Aþ 1 C 1 þ LA1 C 2 B2 þ LA1 Q2 RB2 þ LA1 LK ZRB2 þ LA1 WRH RB2 ;
ð2:11Þ
where W ; Z are arbitrary matrices over H with appropriate sizes. Substituting (2.11) into (1.4) yields þ f ðX Þ ¼ C 4 A4 Aþ 1 C 1 B4 A4 LA1 C 2 B2 B4 A4 LA1 Q2 RB2 B4 A4 LA1 LK ZRB2 B4 A4 LA1 WRH RB2 B4 :
Put þ C 4 A4 Aþ 1 C 1 B4 A4 LA1 C 2 B2 B4 A4 LA1 Q2 RB2 B4 ¼ C;
ð2:12Þ
A4 LA1 LK ¼ D1 ;
ð2:13Þ
A4 LA1 ¼ D2 ;
RB2 B4 ¼ E1 ;
RH RB2 B4 ¼ E2 ;
then f ðX Þ ¼ C D1 ZE1 D2 WE2 :
ð2:14Þ
Note that RðD1 Þ RðD2 Þ, NðE2 Þ NðE1 Þ. Thus, applying (2.7) and (2.8) to (2.14), we get the following: " # " ( #) C C D1 max r½f ðX Þ ¼ max rðC D1 ZE1 D2 WE2 Þ ¼ min r½ C D2 ; r ;r ; ð2:15Þ Z;W A1 X ¼C 1 E1 E2 0 XB ¼C 2
2
A3 XB3 ¼C 3
min r½f ðX Þ ¼ min rðC D1 ZE1 D2 WE2 Þ
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
Z;W
¼ r½ C
C C D2 þ r þr E1 E2
D1 C r 0 E1
Now we simplify the ranks of block matrices in (2.15) and (2.16).
D1 C r 0 E2
D2 : 0
ð2:16Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
737
In view of Lemma 2.3, block Gaussian elimination, (2.3), (2.4), (2.12) and (2.13), we have the following: " # A1 rðKÞ ¼ rðA3 LA1 Þ ¼ r rðA1 Þ; A3 rðH Þ ¼ rðRB2 B3 Þ ¼ r½ B2 B3 rðB2 Þ; C A4 LA1 LK C D1 r ¼r RH RB2 B4 0 E2 0 2 3 C A4 LA1 0 6 7 0 H 5 rðKÞ rðH Þ ¼ r4 RB2 B4 0 K 0 2 3 C A4 LA1 0 6 7 0 RB2 B3 5 rðKÞ rðH Þ ¼ r4 RB2 B4 0 A3 LA1 0 2 3 C A4 0 0 6B 7 6 4 0 B3 B2 7 ¼ r6 7 rðKÞ rðH Þ rðA1 Þ rðB2 Þ 4 0 A3 0 0 5 2
0 A1 0 C4 A4
6 B 6 4 ¼ r6 4 0
0 A3
0 0 B3 C 3
C 1 B4 A1 0 A1 0 0 6 A A C C 3 2 3 6 3 ¼ r6 4 A4 0 0 2
0
3
B2 7 7 7 rðKÞ rðH Þ rðA1 Þ rðB2 Þ A3 C 2 5 0 3 C 1 B4 0 7 A1 7 r ½ B2 7r C4 5 A3
B3 :
ð2:17Þ
0 B2 B3 B4 Similarly, we can obtain the following: C 1 B4 A1 r½ C D2 ¼ r rðA1 Þ; C4 A4 C B2 B4 r ¼r rðB2 Þ; E1 A4 C 2 C 4 2 3 0 C 1 B4 A1 6 A A C C D1 A1 0 7 3 2 6 3 7 r r rðB2 Þ; ¼ r6 7 4 A4 E1 0 0 C4 5 A3
ð2:19Þ
0 B2 B4 3 A1 0 0 C 1 B4 C D2 6 7 C 4 5 r½ B2 B3 rðA1 Þ: r ¼ r 4 A4 0 0 E2 0 0 B2 B 3 B4 Substituting (2.17)–(2.21) into (2.15) and (2.16) yields (2.9) and (2.10). h
ð2:21Þ
ð2:18Þ
ð2:20Þ
2
In Theorem 2.5, letting C4 vanish and A4 and B4 be I and I with appropriate sizes, respectively, we have the following. Corollary 2.6. Assume that A1 2 Hmn , A3 2 Hkn , B2 2 Hrs , B3 2 Hrp , C 1 2 Hmr , C 2 2 Hns , C 3 2 Hkp are given, then the maximal and minimal ranks of the general solution X of the consistent system (1.2) can be expressed as
738
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
max rðX Þ ¼ minfa; b; cg where a ¼ n þ rðC 1 Þ rðA1 Þ; b ¼ r þ rðC 2 Þ rðB2 Þ; A1 C 2 C 1 B3 A1 c¼r r r½ B2 B3 þ n þ r; A3 C 2 C3 A3 A 1 C 2 C 1 B3 A1 C 2 min rðX Þ ¼ r r r½ C 1 B2 C 1 B3 þ rðC 1 Þ þ rðC 2 Þ: A3 C 2 C3 A3 C 2 Remark 2.1. Corollary 2.6 is the main results of [5]. In Theorem 2.5, assuming A1 ; B2 ; C 1 and C2 vanish, we have the following. Corollary 2.7. Suppose that the matrix equation A3 XB3 ¼ C 3 is consistent, then the extremal ranks of the quaternion matrix expression f(X) defined as (1.4) subject to A3 XB3 ¼ C 3 are the following: 9 8 2 3 A3 C 3 0 > > = < B4 6 7 0 C 4 5 rðA3 Þ rðB3 Þ ; max r½f ðX Þ ¼ min r½ C 4 A4 ; r ; r 4 A4 A3 XB3 ¼C 3 > > C4 ; : 0 B3 B4 2 3 2 3 A3 0 A3 C 3 0 A4 0 C 4 B4 6 7 6 7 min r½f ðX Þ ¼ r½ C 4 A4 þ r 0 C 4 5 r 4 A4 C 4 5 r þ r 4 A4 : A3 XB3 ¼C 3 C4 0 B 3 B4 0 B3 B4 0 B4 Remark 2.2. Corollary 2.7 is Theorem 3.2 in [8]. 3. A practical solvability condition of system (1.3) In this section, we use Theorem 2.5 to give a necessary and sufficient condition for the consistency to the system (1.3) by rank equalities. Theorem 3.1. Let A1 2 Hmn , C 1 2 Hmr , B2 2 Hrs , C 2 2 Hns , A3 2 Hkn , B3 2 Hrp , C 3 2 Hkp , A4 2 Hqn , B4 2 Hrl , C 4 2 Hql be known and X 2 Hnr unknown, then the system (1.3) is consistent if and only if (2.2), (2.5) and (2.6) hold, and the following equalities are all satisfied: B4 r½ A4 C 4 ¼ rðA4 Þ; r ð3:1Þ ¼ rðB4 Þ; C4 B4 B2 ð3:2Þ r ¼ r½ B2 B4 ; A4 C 2 C 4 C 1 B4 A1 A1 r ¼r ; ð3:3Þ C4 A4 A4 2 3 2 3 0 0 C 1 B4 A1 A1 6 A A C C 7 0 7 3 2 3 6 3 6 7 ð3:4Þ r6 7 ¼ r4 A3 5 þ r½ B2 B3 B4 : 4 A4 0 0 C4 5 A4 0 B2 B3 B4 Proof. It is obvious that the system (1.3) is consistent if and only if the system (1.2) is consistent and min r½f ðX Þ ¼ 0;
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
ð3:5Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
739
where f(X) is defined as (1.4). Let X0 be a solution to the system (1.3). Then X0 is a solution to system (1.2) and X0 satisfies A4 X 0 B4 ¼ C 4 . Hence, Lemma 2.2 yields (2.2), (2.5), (2.6) and (3.1). It follows from 2
0 0
I
6 60 6 60 4 0
I
0
0
I
0
32
A1
76 A3 X 0 7 6 A3 76 6 0 7 5 4 A4 I 0
0 0
0
0
A3 C 2
C 3
0
0
B2
B3
C 1 B4
32
I
76 0 76 0 76 6 C4 7 54 0 0 B4
0
0 X 0 B4
I
0
0
0
I
0
0
0
I
3
2
A1
7 6 7 6 A3 7¼6 7 6A 5 4 4 0
0
0
0
0
0
0
B2
B3
0
3
7 0 7 7 0 7 5 B4
that (3.4) holds. Similarly, we can obtain (3.2) and (3.3). Conversely, assume that (2.2), (2.5) and (2.6) hold, then by Lemma 2.2, system (1.2) is consistent. By (2.10), (3.2)–(3.4), and noting that 2
0
A1
6A 6 3 r6 4 A4
A3 C 2 0
0 A1 6 r 4 A4
B2 0 0
0 0
0
B2
B3
2
3
2 3 A1 7 0 7 6 7 7 P r4 A3 5 þ r½ B2 B4 ; C4 5 A4 B4 3 C 1 B4 A1 7 C4 5 P r þ r ½ B2 B3 B4 A4 B4
C 1 B4
yields min r½f ðX Þ 6 0:
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
However, min r½f ðX Þ P 0:
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
Hence (3.5) holds, implying the system (1.3) is consistent.
h
In Theorem 3.1 and its proof, let B2 ; C 2 vanish and A1 C1 A1 ¼ ; C1 ¼ : A2 C2 Then we can get the following. Corollary 3.2. The system of quaternion matrix equations (1.5) is consistent if and only if (2.5), (3.1) hold and the following equalities are all satisfied: " # " # A1 C 1 A1 r ¼r ; A2 C 2 A2 2 3 2 3 A1 A1 C 1 B 3 6 7 6 7 r 4 A2 C 2 B 3 5 ¼ r 4 A2 5 ; 2
A3 C 1 B4
6 r 4 C 2 B4 C4
C3 A1
3
2
A3 A1
3
7 6 7 A 2 5 ¼ r 4 A2 5 ; A4
A4
740
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
2
A1
6 6 A2 6 r6 6 A3 6 4 A4 0
0 0 C 3 0 B3
C 1 B4
3
2 3 A1 7 C 2 B4 7 6A 7 7 6 27 0 7 7 þ r ½ B3 7 ¼ r6 4 A3 5 7 C4 5 A4 B4
B4 :
Remark 3.1. Corollary 3.2 gives a practical solvability condition to system (2.5) which was investigated in [25]. In Theorem 3.1 and its proof, letting A1 ; C 1 vanish and putting B2 ¼ ½ B1 following.
B2 and C 2 ¼ ½ C 1
C 2 yields the
Corollary 3.3. The system of quaternion matrix equations (1.6) has a solution if and only if (2.5), (3.1) hold and the following equalities are all satisfied: B1 B 2 r ¼ r½ B1 B2 ; C1 C2 B1 B2 B3 r ¼ r½ B1 B2 B3 ; A3 C 1 A3 C 2 C 3 B1 B2 B4 r ¼ r½ B1 B2 B4 ; A4 C 1 A4 C 2 C 4 2 3 A3 A3 C 1 A3 C 2 C 3 0 A3 6 7 r 4 A4 0 0 0 C4 5 ¼ r þ r½ B1 B2 B3 B4 : A4 0 B1 B2 B3 B4 Remark 3.2. Corollary 3.3 presents a practical necessary and sufficient condition for the solvability to system (1.6) which was studied in [31]. Corollary 3.4 ([29,30]). Suppose that A3 ; B3 ; C 3 ; A4 ; B4 and C4 are those in Theorem 3.1, then the system of quaternion matrix equations A3 XB3 ¼ C 3 , A4 XB4 ¼ C 4 is consistent if and only if (2.5), (3.1) hold and the following equality is satisfied: 2 3 2 3 A3 C 3 0 A3 0 0 6 7 6 7 r 4 A4 0 C 4 5 ¼ r4 A4 0 0 5: 0
B3
B4
0
B3
B4
By Theorem 3.1, we can also get the following. Corollary 3.5. Let A1 2 Hmn , C 1 2 Hmr , B2 2 Hrs , C 2 2 Hns and A; B 2 Hrn . Then A and B have a common inner inverse which is a solution of the system (1.1) if and only if (2.2) holds and the following equalities are all satisfied: B2 A r ¼ r½ B2 A ; AC 2 A B2 B r ¼ r½ B2 B ; BC 2 B A1 C 1 A A1 r ; ¼r A A A C 1 B A1 A1 r ¼r ; B B B
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
2
A1
6A 6 r6 4B 0
0
0
AC 2 0
A 0
B2
A
3
2 3 A1 7 0 7 6 7 ¼ r 7 4 A 5 þ r ½ B2 B 5 B B
C1B
741
A
B :
4. Extremal ranks of schur complement subject to (1.1) As is well known, for a given block matrix A B M¼ ; C D where A; B; C and D are quaternion matrices with appropriate sizes, a generalized Schur complement of A in M is defined as S A ¼ D CA B;
ð4:1Þ
where A is an inner inverse of A, i.e. A 2 fX j AXA ¼ Ag. Now we use Theorem 2.5 to establish the extremal ranks of SA given by (4.1) with respect to A which is a solution to system (1.1). Theorem 4.1. Suppose A1 2 Hmn , C 1 2 Hmr , B2 2 Hrs , C 2 2 Hns , D 2 Hql , C 2 Hqn , B 2 Hrl , A 2 Hrn are given and system (1.1) is consistent, then the extreme ranks of SA given by (4.1) with respect to A which is a solution of (1.1) are the following: max rðS A Þ ¼ minfa; b; cg;
A1 A ¼C 1 A B2 ¼C 2
where a¼r
C1B
D B2
A1
C B
rðA1 Þ;
rðB2 Þ; CC 2 D 2 3 A1 0 0 C1B 6 A AC A A1 0 7 2 6 7 r c ¼ r6 r½ B2 A ; 7 4C 0 0 D 5 A 0 B2 A B 2 0 0 A1 6 A AC A C 1 B A1 B2 B 2 6 min rðS A Þ ¼ r þr þ r6 A1 A ¼C 1 4 CC C 0 0 D C D 2 A B2 ¼C 2 0 B2 A 2 3 2 3 A1 0 C1B A1 0 0 C 1 B 6 A AC 0 7 2 6 7 6 7 r6 D 5: 7 r4 C 0 0 4C 0 D 5 0 B2 A B 0 B2 B b¼r
Proof. It is obvious that max rðD CA BÞ ¼ max rðD CXBÞ;
A1 A ¼C 1 A B2 ¼C 2
A1 X ¼C 1 XB2 ¼C 2 AXA¼A
C1B
3
0 7 7 7 D 5 B
742
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
min rðD CA BÞ ¼ min rðD CXBÞ:
A1 A ¼C 1 A B2 ¼C 2
A1 X ¼C 1 XB2 ¼C 2 AXA¼A
Thus in Theorem 2.5 and its proof, letting A3 ¼ B3 ¼ C 3 ¼ A, A4 ¼ C, B4 ¼ B, C 4 ¼ D, we can easily get the proof. h In Theorem 4.1, let A1 ; C 1 ; B2 ; C 2 vanish. Then we can easily get the following. Corollary 4.2. The extreme ranks of SA given by (4.1) with respect to A are the following: ( " # " # ) B A B max rðS A Þ ¼ min r½ C D ; r ;r rðAÞ ; A D C D 2 3 A 0 " # " # " # B A B A 0 B 6 7 7 min rðS Þ ¼ rðAÞ þ r½ C D þ r þ r r r6 A 4 0 B 5: A D C D 0 C D C D Remark 4.1. Corollary 4.2 is Theorem 4.1 in [8]. 5. The rank invariance of (1.4) As another application of Theorem 2.5, we in this section consider the rank invariance of the matrix expression (1.4) subject to the consistent system (1.2). Theorem 5.1. Suppose that (1.2) is consistent, then the rank of f(X) defined by (1.4) subject to (1.2) is invariant if and only if 2
0
A1
6 6 A3 6 r6 6 A4 4
A3 C 2 0
0 2
or
B2 0
C 1 B4
0
0
C4
0
B2
B3
A1 6 r 4 A4
0 0
0 0
0 A1 6A 6 3 r6 4 A4
B2
0
B3 0 A3 C 2 0 B2
B4
#
" þr
C4
A1
# ;
A3
B4
0
2
3
7 " B2 0 7 7 7¼r C4 7 A4 C 2 5
A1 6 r6 4 A4
2
or
C 1 B4
B4
3
2
A1
6 " # 6 A3 A1 7 6 7þr ¼ r6 5 6 A4 A3 4 0
0
0
A3 C 2
C 3
0
0
B2
B3
C 1 B4
3 C 1 B4 C 1 B4 A1 7 C4 5 ¼ r þ r½ B2 B3 ; C4 A4 B4 3 2 A1 C 1 B4 0 0 7 6 0 7 6 A3 A3 C 2 C 3 7 þ r ½ B2 B3 ¼ r 6 4 A4 C4 5 0 0 B4 0 B2 B3
3
7 0 7 7 7 þ rðA1 Þ C4 7 5
ð5:1Þ
B4
3 C 1 B4 0 7 7 7 þ rðB2 Þ C4 5 B4
ð5:2Þ
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
2
0
A1
6 6 A3 r6 6A 4 4
A3 C 2 0
0 2
C 1 B4
B2
3
" 7 B2 0 7 7¼r 7 C4 5 A4 C 2 B4
A1
0
0
C 1 B4
6 r 4 A4
0
0
C4
0
B2
B3
B4
743
B4
# þr
C4
3
" C 1 B4 7 5¼r C4
A1
"
A1 A3
# ;
#
A4
þ r ½ B2
B3 :
ð5:3Þ
Proof. It is obvious that the rank of f(X) subject to the consistent system (1.2) is invariant if and only if max r½f ðX Þ min r½f ðX Þ ¼ 0:
ð5:4Þ
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
A1 X ¼C 1 XB2 ¼C 2 A3 XB3 ¼C 3
By (5.4), Theorem 2.5 and simplifications, we can get (5.1), (5.2) and (5.3). h Corollary 5.2. The rank of SA defined by (4.1) with respect to A which is a solution to system (1.1) is invariant if and only if 2
6A 6 r6 4C 0 2
or
0
C1B
AC 2 0
0 D
B2
B
A1
A1 6 r4 C
0 0
0 0
0
B2
A
A1
0
0
2
6 r4 C 0 2 A1 6A 6 r6 4C
3 7 B2 7 7¼r 5 CC 2
B A1 þr ; D A
2 3 A1 C1B 6 A1 6A 7 D 5þr ¼ r6 4C A B 0
C1B
0 AC 2
0 A
0
0
B2
A
B2
B
3
A1
0
C1B
6A 6 r6 4C
AC 2 0
0 D
0 A1
B2 0 0
0
B2
or 2
2
6 r4 C 0
B
C 1 B A1 7 0 0 D 5¼r þ r½ B2 A ; D C B2 A B 3 2 A1 0 C1B 0 0 7 6 AC 2 0 7 6 A AC 2 A 7 þ r½ B2 A ¼ r6 4C 0 D 5 0 0
0
0 B2
0 A
3 C1B 0 7 7 7 þ rðA1 Þ D 5
3 7 B2 7 7¼r 5 CC 2
B 3 C1B C1B 7 D 5¼r D B
B A1 þr ; D A A1 þ r½ B2 C
A :
A
C1B 0 D B
3 7 7 7 þ rðB2 Þ 5
744
Q.-W. Wang et al. / Applied Mathematics and Computation 195 (2008) 733–744
6. Conclusion In this paper, we have derived the maximal and minimal ranks of the quaternion matrix expression C 4 A4 XB4 subject to the consistent system of quaternion matrix equations A1 X 1 ¼ C 1 , X 1 B2 ¼ C 2 , A3 XB3 ¼ C 3 . Using the results, we give new and practical solvability conditions to systems of quaternion matrix Eqs. (1.3), (1.5) and (1.6). Moreover, maximal and minimal ranks of the generalized Schur complement D CA B with respect to an inner inverse A of A, which is a common solution to matrix equations A1 X ¼ C 1 ; XB2 ¼ C 2 , are also investigated. Some known results can be viewed as the special cases of this paper. References [1] T.W. Hungerford, Algebra, Spring-Verlag, New York Inc., 1980. [2] S.K. Mitra, The matrix equations AX ¼ C, XB ¼ D, Linear Algebra Appl. 59 (1984) 171–181. [3] P. Bhimasankaram, Common solutions to the linear matrix equations AX ¼ B; XC ¼ D, and EXF ¼ G, Sankhya Ser. A 38 (1976) 404–409. [4] C.Y. Lin, Q.W. Wang, New solvability conditions and a new expression of the general solution to a system of linear matrix equations over an arbitrary division ring, Southeast Asian Bull. Math. 29 (5) (2005) 755–762. [5] C.Y. Lin, Q.W. Wang, The minimal and maximal ranks of the general solution to a system of matrix equations over an arbitrary division ring, Math. Sci. Res. J. 10 (3) (2006) 57–65. [6] Q.W. Wang, A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin. 21 (2) (2005) 323–334. [7] Y. Tian, S. Cheng, The maximal and minimal ranks of A BXC with applications, New York J. Math. 9 (2003) 345–362. [8] Y. Tian, Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear Algebra Appl. 355 (2002) 187– 214. [9] T. Ando, Generalized Schur complements, Linear Algebra Appl. 27 (1979) 173–186. [10] M. Fiedler, Remarks on the Schur complement, Linear Algebra Appl. 39 (1981) 189–195. [11] D. Carlson, E. Haynsworth, T. Markham, A generalization of the Schur complements by means of the Moore-Penrose inverse, SIAM J. Appl. Math. 26 (1974) 169–179. [12] D. Carlson, What are Schur complements, anyway? Linear Algebra Appl. 74 (1986) 257–275. [13] Y. Tian, More on maximal and minimal ranks of Schur complements with applications, Appl. Math. Comput. 152 (3) (2004) 675–692. [14] S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, Oxford, 1995. [15] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997) 21–57. [16] A. Baker, Right eigenvalues for quaternionic matrices: a topological approach, Linear Algebra Appl. 286 (1999) 303–309. [17] D.I. Merino, V. Sergeichuk, Littlewood’s algorithm and quaternion matrices, Linear Algebra Appl. 298 (1999) 193–208. [18] S. De Leo, G. Scolarici, Right eigenvalue equation in quaternionic quantum mechanics, J. Phys. A 33 (2000) 2971–2995. [19] D.R. Farenick, B.A.F. Pidkowich, The spectral theorem in quaternions, Linear Algebra Appl. 371 (2003) 75–102. [20] C.E. Moxey, S.J. Sangwine, T.A. Ell, Hypercomplex correlation techniques for vector images, IEEE Trans. Signal Process. 51 (7) (2003) 1941–1953. [21] N. Le Bihan, J. Mars, Singular value decomposition of matrices of quaternions: a new tool for vector-sensor signal processing, Signal Process. 84 (7) (2004) 1177–1199. [22] N. Le Bihan, S.J. Sangwine, Quaternion principal component analysis of color images, in: IEEE International Conference on Image Processing (ICIP), Barcelona, Spain, September, 2003. [23] N. Le Bihan, S.J. Sangwine, Color image decomposition using quaternion singular value decomposition, in: IEEE International Conference on Visual Information Engineering (VIE), Guildford, UK, July, 2003. [24] S.J. Sangwine, N. Le Bihan, Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations, Appl. Math. Comput. 182 (1) (2006) 727–738. [25] Q.W. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 665–675. [26] Q.W. Wang, Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 641–650. [27] Q.W. Wang, G.J. Song, C.Y. Lin, Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application, Appl. Math.Comput. 189 (2007) 1517–1532. [28] Q.W. Wang, Z.C. Wu, C.Y. Lin, Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Appl. Math. Comput. 182 (2006) 1755–1764. [29] Q.W. Wang, A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra Appl. 384 (2004) 43–54. [30] S.K. Mitra, A pair of simultaneous linear matrix equations A1 XB1 ¼ C 1 and A2 XB2 ¼ C 2 and a programming problem, Linear Algebra Appl. 131 (1990) 107–123. [31] Q.W. Wang, F. Qin, C.Y. Lin, The common solution to matrix equations over a regular ring with applications, Indian J. Pure Appl. Math. 36 (12) (2005) 655–672.