A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity

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1 Linear Algebra and its Applications ) A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity Qing-Wen Wang Department of Mathematics, Shanghai University, Shanghai , People s Republic of China Received 22 May 2003; accepted 30 December 2003 Submitted by R. Guralnick Abstract In this paper we consider the classical system of matrix equations { A1 XB 1 = C 1, A 2 XB 2 = C 2 over R, an arbitrary regular ring with identity. Necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived. As an application, the linear matrix equation AXB CYD = E over R is considered Published by Elsevier Inc. AMS classification: 15A06; 15A24; 15A33; 15A09; 16E50 Keywords: Regular ring; Linear matrix equation; System of matrix equations; Inner inverse of a matrix; Reflexive inverse of a matrix 1. Introduction Since Mitra [1] first studied the system { A1 XB 1 = C 1, A 2 XB 2 = C 2 1.1) Supported by a grant of Development Foundation of Educational Committee of Shanghai and grants of Natural Science Foundation of China and Shanghai. address: wqw858@ustc.edu Q.-W. Wang) /$ - see front matter 2004 Published by Elsevier Inc. doi: /j.laa

2 44 Q.-W. Wang / Linear Algebra and its Applications ) over the complex field in 1973, there have been many papers to discuss the system e.g. [1 8]). For instance, Van der Woude [4,5] investigated it over a field in Özgüler and Akar [6] gave a condition for the solvability of the system over a principle domain in Wang [8] studied the system over an arbitrary division ring in Many problems in systems and control theory require the solution of generalized Sylvester s matrix equation AX YB = C. Roth [9] gave a necessary and sufficient condition for the consistency of the matrix equation, which was called Roth s equivalence theorem. Since Roth s paper appeared in 1952, generalized Sylvester s matrix equation has been widely studied e.g. [10 31]). In 1976, Hartwig [10] generalized Roth s equivalence theorem to a unite regular ring. The matrix equation AXB CYD = E 1.2) is a further generalization of generalized Sylvester s equation, which was investigated by many authors, such as Baksalary and Kala [28], Özgüler [29], Huang et al. [30,31], Wang [8], and the others. In this paper we consider the system 1.1) over R. InSection2,wederivesome necessary and sufficient conditions for the existence and a representation of the general solution to the system 1.1). As an application of Section 2, we in Section 3 give necessary and sufficient conditions for the existence and the expressions of the general solution to the matrix equation 1.2) over R. We present some brief comments and further research problems with respect to the system 1.1) in Section 4. A ring R is called regular if for every a R there exists a 1) R such that aa 1) a = a. a 1) is termed with an inner inverse of a. Throughout we denote a regular ring with identity 1 by R,thesetofallm n matrices over R by R m n, the identity matrix with the appropriate size by I, an inner inverse of a matrix A over R by A 1) which satisfies AA 1) A = A, a reflexive inverse of a matrix A over R by A which satisfies simultaneously AA A = A and A AA = A. Clearly, if X is an inner inverse, then XAX is a reflexive inverse of A. So a matrix A has an inner inverse if and only if A has a reflexive inverse. Moreover, L A = I A A, R A = I AA where A is arbitrary but fixed. Let {[ ] } A C T mr,sn = A R m s,b R r n,c R m n. 0 B For matrices A 1 R m s, B 1 R r n and C 1 R m n, if there exist invertible matrices P T mr,mr and Q T sn,sn such that [ ] [ ] A1 C 1 A1 0 = P Q, 0 B 1 0 B 1 then we say that [ ] [ ] A1 C 1 A B 1 0 B 1

3 Q.-W. Wang / Linear Algebra and its Applications ) By [32], we have the following. Proposition 1.1. The full ring of square matrices over a regular ring is regular. Proposition 1.2. reflexive inverse. AringR is regular if and only if every matrix of R m n has a Proof. Let every matrix of R m n have a reflexive inverse. Then for an arbitrary a R, anda = ae 11 R m n where E 11 is the matrix with 1 as the 1, 1) entry and 0 elsewhere, we can assume that A 1) = b ij ) n m. It follows from ae 11 = ae 11 )A 1) ae 11 ) that a = ab 11 a. This implies R is regular. Conversely, let R be a regular ring. Then for any matrix A R m n, by Proposition 1.1, A has an inner inverse, thereby a reflexive inverse if m = n. For the case where m/= n, without loss of generality we can assume m>n. Augment A by zero columns so that A, 0) R m m. Then by Proposition 1.1, A, 0) has an inner inverse termed with B = [ B 1 ] B 2 where B1 R n m. Thus it follows from: [ ] B1 A, 0) A, 0) = A, 0) B 2 that AB 1 A = A, i.e., A 1) = B 1. Hence A has a reflexive inverse B 1 AB The general solution to the system 1.1) over R In 1976, Hartwig [10] gave the fact that Roth s equivalence theorem does not hold over a ring with elements a, b such that ba = 1 /= ab. It follows from the fact that Roth s equivalence theorem need not hold over R. However, by Proposition 1.2 and [12], we have the following. Lemma 2.1. Let A R m s,b R r n and C R m n. Then the following conditions are equivalent: 1) The matrix equation AX YB = C over R is consistent. 2) R A CL B = 0. 3) [ ] A C 0 B [ ] A 0. 0 B It is easy to prove the following: Lemma 2.2. Let A R m s,b R r n and C R m n. Then the following statements are equivalent:

4 46 Q.-W. Wang / Linear Algebra and its Applications ) ) The matrix equation AXB = C 2.1) is consistent. 2) [ ] [ ] [ ] [ ] A C A 0 0 C 0 0, B 0 B 3) AA CB B = C. 4) AA C = C, CB B = C. In that case, the general solution of the matrix equation 2.1) is X = A CB L A V UR B where U, V are any matrices with compatible dimensions over R. Lemma 2.3. Let A, B be matrices over R and [ ] A1 A =, A 2 B =[B 1 B 2 ], S = A 2 L A1, T = R B1 B 2. Then A = [ A 1 L A 1 S A 2 A 1 L A1 S ], [ B B = 1 B 1 B 2T ] R B1 T R B1 2.2) are a reflexive inverse of A and B, respectively. Proof. Note that A 1 L A1 = 0, R B1 B 1 = 0. Then it can be verified immediately that 2.2) holds by the definition. Now we give the main theorem of this paper. Theorem 2.4. Let A 1 R m n,a 2 R s n,b 1 R r p,b 2 R r t,c 1 R m p, C 2 R s t be known matrices and X R n r unknown, S= A 2 L A1,T = R B1 B 2, F = B 2 L T,G= R S A 2. Then the following conditions are equivalent: 1) The system 1.1) is consistent. 2) A i A i C ib i B i = C i, i = 1, 2 2.3) and G A 2 C 2B 2 A 1 C 1B ) 1 F = )

5 Q.-W. Wang / Linear Algebra and its Applications ) ) A 1 C 1 0 A A 2 0 C 2 A ) 0 B 1 B 2 0 B 1 B 2 and [Ai C i 0 0 ] [ ] Ai 0, 0 0 [ ] [ ] 0 Ci 0 0, i = 1, ) 0 B i 0 B i In that case, the general solution of the system 1.1) can be expressed as the following X = A 1 C 1B 1 L A 1 S A 2 L G A 2 C 2B 2 A 1 C 1B ) 1 B2 B 2 G G A 2 C 2B 2 A 1 C 1B ) 1 B2 T R B1 L A1 Y S SYB 2 B ) 2 LA1 S A 2 L G WTB 2 W G GWT T ) R B1, 2.7) where Y and W are any matrices over R with appropriate dimensions. Proof. 1) 2): Let the system 1.1) have a solution X 0.ThenA i X 0 B i = C i, i = 1, 2. It follows from Lemma 2.2 that 2.3) holds and X 0 = A 1 C 1B 1 L A 1 V UR B1, where U and V are matrices over R. Hence by A 2 X 0 B 2 = C 2, A 2 UT SVB 2 = C 2 A 2 A 1 C 1B 1 B ) Note that R S S = 0, TL T = 0. Thus by 2.3) and 2.8), G A 2 C 2B 2 A 1 C 1B ) 1 F = RS C2 A 2 A 1 C 1B 1 B ) 2 LT = R S A 2 UT SVB 2 )L T = 0, i.e., 2.4) holds. 2) 1): Suppose that 2.3) and 2.4) hold. Note that 2.4) yields G A 2 C 2B 2 A 1 C 1B ) 1 B2 T T = G A 2 C 2B 2 A 1 C 1B ) 1 B2. 2.9) By A 1 L A1 = 0, R B1 B 1 = 0 and 2.3), it can be verified that the matrix X that has the form of 2.7) is a solution of A 1 XB 1 = C 1. Now we show that the matrix X that has the form of 2.7) is also a solution of A 2 XB 2 = C 2. By virtue of SS A 2 = A 2 G, TB 2 B 2 = T, 2.3) and 2.9), we have the following A 2 XB 2 = A 2 A 1 C 1B 1 B 2 A 2 L A1 S A 2 L G A 2 C 2B 2 A 1 C 1B ) 1 B2 B 2 B 2 A 2 G G A 2 C 2B 2 A 1 C 1B ) 1 B2 T R B1 B 2 A 2 L A1 Y S SYB 2 B ) 2 B2 A 2 L A1 S A 2 L G WTB 2 B 2 A 2 W G GWT T ) R B1 B 2

6 48 Q.-W. Wang / Linear Algebra and its Applications ) ) B2 = A 2 A 1 C 1B 1 B 2 A 2 G)L G A 2 C 2B 2 A 1 C 1B 1 A 2 G G A 2 C 2B 2 A 1 C 1B ) 1 B2 S Y S SYB 2 B ) 2 B2 A 2 G)L G WT A 2 W G GWT T ) T = A 2 A 1 C 1B 1 B 2 A 2 A 2 C 2B 2 A 1 C 1B ) 1 B2 = C ) So the matrix X has the form of 2.7) is a solution of the system of 1.1). 1) 3): Let the system 1.1) have a solution X 0.ThenA 1 X 0 B 1 =C 1, A 2 X 0 B 2 = C 2. Accordingly, 2.6) holds by Lemma 2.2, and it follows from I 0 0 A 1 C 1 0 I X 0 B I A 2 X 0 A 2 0 C 2 0 I I 0 B 1 B I A = A B 1 B 2 that 2.5) holds. 3) 2): If 2.6) holds, then 2.3) holds by Lemma 2.2. Now suppose that 2.5) holds, then it follows from Lemma 2.1 that [ [ ][ ] ] [C1 ] A1 A1 0 [ I I [ ] [ ] ] B A 2 A 2 0 C 1 B 2 B1 B 2 = ) By Lemma 2.3 and 2.3), [ ][ ] [ ] A1 A1 RA1 0 I = A 2 A 2 GA, 1 R S [ I [ ] [ ] ] [ LB1 B B 1 B 2 B1 B 2 = 1 F ]. 0 L T Hence 2.11) yields [ RA1 C 1 L B1 R A1 C 1 B 1 F ] GA 1 C 1L B1 GA 1 C 1B 1 A 2 C 2B 2 )F = 0. Therefore GA 1 C 1B 1 A 2 C 2B 2 )F = 0, i.e. 2.4) holds. Now we show that if the system 1.1) is consistent, i.e., 2.3) and 2.4) hold, then its general solution can be expressed as 2.7). In 2) 1), wehaveshownthatthe matrix X that has the form of 2.7) is a solution of the system 1.1). So we only need to prove that for an arbitrary solution X 0 of the system 1.1) can be expressed as the form of 2.7).

7 Q.-W. Wang / Linear Algebra and its Applications ) Let U = W G GWT T G G A 2 C 2B 2 A 1 C 1B ) 1 B2 T, V = Y S SYB 2 B 2 S C 2 A 2 A 1 C 1B 1 B 2 A 2 UT ) B 2. Then by 2.9) and 2.3), 2.7) becomes X = A 1 C 1B 1 L A 1 V UR B ) Suppose that W = X 0, Y = X 0 B 1 B 1 where X 0 is an arbitrary solution of the system 1.1). Then by 2.3), X 0 U = G GX 0 TT G G A 2 C 2B 2 A 1 C 1B ) 1 B2 T = G R S A2 X 0 R B1 B 2 C 2 A 2 A 1 C 1B 1 B ) 2 T = G R S A2 A 1 C 1B 1 B 2 A 2 X 0 B 1 B 1 B ) 2 T = G R S A2 A 1 A 1X 0 B 1 B 1 B 2 A 2 X 0 B 1 B 1 B ) 2 T = G R S A 2 L A1 X 0 B 1 B 1 B 2T i.e., X 0 = U. So = G R S SX 0 B 1 B 1 B 2T = 0, X 0 B 1 B 1 V = S SX 0 B 1 B 1 B 2B 2 S C 2 A 2 A 1 C 1B 1 B 2 A 2 X 0 T ) B 2 = S SX 0 B 1 B 1 B 2 C 2 A 2 A 1 C 1B 1 B 2 A 2 X 0 T ) B 2 = S A 2 X 0 B 1 B 1 B 2 A 2 A 1 A 1X 0 B 1 B 1 B 2 C 2 A 2 A 1 C 1B 1 B 2 A 2 X 0 T ) B 2 = S [ A 2 X 0 B1 B 1 R B 1 ) B2 C 2 ] B 2 = S A 2 X 0 B 2 C 2 ) B 2 = 0, i.e., X 0 B 1 B 1 = V. Hence X 0 = A 1 C 1B 1 L A 1 X 0 B 1 B 1 X 0R B1 can be expressed as 2.12), i.e., 2.7) where W = X 0, Y = X 0 B 1 B The linear matrix equation 1.2) over R In this section, using Theorem 2.4, we consider the linear matrix equation 1.2) where A R m n, B R p q, C R m r, D R l q, E R m q are known. Let

8 50 Q.-W. Wang / Linear Algebra and its Applications ) M = R A C, N = DL B, S = CL M, T = R D N, F = NL T, G = R S C, P = R C A, Q = BL D, S 1 = AL P, T 1 = R B Q,andG 1 = R S1 A. Then we have the following. Theorem 3.1. equivalent: For the linear matrix equation 1.2), the following statements are 1) 1.2) is consistent. 2) R M R A E = 0, R A EL D = 0, EL B L N = 0, R C EL B = 0. 3) MM R A ED D = R A E, CC EL B N N = EL B. 4) R P R C E = 0, R C EL B = 0,R A EL D = 0, EL D L Q = 0. 5) PP R C EB B = R C E, AA EL D Q Q = EL D. 6) [ ] [ ] A E A 0, 3.1) 0 D 0 D [ ] [ ] C E C 0, 3.2) 0 B 0 B [ ] [ ] A C) E A C) 0, 3.3) E 0 0 B B. 3.4) 0 0 D) D) In that case, the general solution of 1.2) can be expressed as the following X = A E CYD)B L A U ZR B, Y = M R A ED L M V S SVNN ) 3.5) L M S CL G WTN W G GWT T )R D, where U, V, W, Z are arbitrary matrices over R with appropriate sizes; or X = P R C EB L P V1 S 1 S 1V 1 QQ ) L P S 1 AL G 1 W 1 T 1 Q W 1 G 1 G 1W 1 T 1 T ) 1 RB, 3.6) Y = C E CXD)D L C U 1 Z 1 R D where U 1,V 1,W 1,Z 1 are arbitrary matrices over R with appropriate sizes. Proof. Clearly, 2) 3), 4) 5). Nowweshowthat1) 2): Obviously, the matrix equation 1.2) is consistent if and only if the matrix equation AXB = E CYD is consistent. By Lemma 2.2, 1.2) is consistent if and only if the system { AA E CYD) = E CYD, E CYD)B B = E CYD,

9 Q.-W. Wang / Linear Algebra and its Applications ) i.e., { MYD = RA E, 3.7) CYN = EL B is consistent. In view of Theorem 2.4, the system 3.7) is consistent if and only if 3) and G C EL B N M R A ED ) F = 0 3.8) hold. Now we prove that if 3), i.e., 2) holds, then G C EL B N M R A ED ) N = 0 3.9) holds, yielding 3.8) holds. Suppose that 2) holds, then R A E = R A ED D, CC EL B = EL B = EL B N N. It follows from S = CL M that CM M = C S. Noting that R S S = 0, we have GM R A ED N = R S CM R A ED DL B = R S CM R A EL B = R S CM R A CC EL B = R S CM MC EL B = R S C S)C EL B = R S CC EL B = R S EL B, GC EL B N N = R S CC EL B N N = R S EL B. Hence, 3.9) holds, yielding 3.8) holds. Therefor 3.7) is consistent, thus 1.2) is consistent if and only if 2) holds. Similarly, we can prove that 1) 4). 1) 6): Assume that 1.2) is consistent, then the matrix equation [ ] [ ][ ] X 0 B A C = E 0 Y D is consistent and by Lemma 2.1, 3.1) and 3.2) hold. Thus by Lemma 2.2, 3.3) and 3.4) hold. Consequently 6) holds. Conversely, suppose that 6) holds, then by 3.3), 3.4) and Lemma 2.1, the matrix equations [ ] A C X = E 3.10) and [ B Y = E 3.11) D] are consistent. So we can suppose X 0, Y 0 are a solution of 3.10) and 3.11), respectively. It follows from DL B L N = NL N = 0, R M M = 0, R A A = 0, BL B = 0 that

10 52 Q.-W. Wang / Linear Algebra and its Applications ) [ ] R M R A E = R M R A A C X0 = [ ] R M R A AX 0 R M R A CX 0 = [ ] R M R A AX 0 R M MX 0 = 0, [ ] [ ] B BLB L EL B L N = Y 0 L D B L N = Y N 0 = 0. DL B L N By 3.1), 3.2) and Lemma 2.1, AX YD = E and CX YB = E are consistent. Hence Lemma 2.1 yields R A EL D = 0, R C EL B = 0, i.e., 2) holds. Therefore 1.2) is consistent. If 1.2) has a solution, then by 3.8), Lemma 2.2 and Theorem 2.4, 3.5) or 3.6) is the general solution of 1.2). 4. Conclusion We have derived necessary and sufficient conditions for the existence and the expression of the general solutions to the system 1.1) over R. The solvability conditions and the representation of the general solution are more straightforward than those over fields given by Mitra [1], Shinozaki and Sibuya [3], Von der Woude [4], Navarra et al. [7]. In particular, Mitra [1,2] requires general solutions of multiple auxiliary equations in his expressions of the general solution to the system 1.1). Our expression of the general solution to the system 1.1) requires no solutions to auxiliary equations and no other tools except for reflexive inverses of matrices. We have utilized our results to give necessary and sufficient conditions for the existence and the expressions of the general solution of the matrix equation 1.2) over R. Using the results of this paper, we can also study selfconjugate solution, centroselfconjugate solution, perselfconjugate solution and their expressions to the matrix equation 2.1) over R with an involutorial antiautomorphism and char R /= 2; the least squares solutions to the system 1.1) and the matrix equation 2.1) over the real quaternion field; investigate the maximal and minimal ranks of the solution to the system 1.1) over an arbitrary division ring. Acknowledgement The author would like to thank an anonymous referee for useful suggestions which helped to improve the exposition of this paper. References [1] S.K. Mitra, A pair of simultaneous linear matrix equations A 1 XB 1 = C 1 and A 2 XB 2 = C 2, Proc. Cambridge Philos. Soc )

11 Q.-W. Wang / Linear Algebra and its Applications ) [2] S.K. Mitra, A pair of simultaneous linear matrix equations and a matrix programming problem, Linear Algebra Appl ) [3] N. Shinozaki, M. Sibuya, Consistency of a pair of matrix equations with an application, Keio Eng. Rep ) [4] J. Van der Woulde, Feedback decoupling and stabilization for linear system with multiple exogenous variables, Ph.D. Thesis, Technical University of Eindhoven, Netherlands, [5] J. Van der Woulde, Almost noninteracting control by measurement feedback, Systems Control Lett ) [6] A.B. Özgüler, N. Akar, A common solution to a pair of linear matrix equations over a principle domain, Linear Algebra Appl ) [7] A. Navarra, P.L. Odell, D.M. Young, A representation of the general common solution to the matrix equations A 1 XB 1 = C 1 and A 2 XB 2 = C 2 with applications, Comput. Math. Appl ) [8] Q.W. Wang, The decomposition of pairwise matrices and matrix equations over an arbitrary skew field, Acta Math. Sinica 39 3) 1996) in Chinese). [9] W.E. Roth, The equation AX YB = C and AX XB = C in matrices, Proc. Amer. Math. Soc ) [10] R.E. Hartwig, Roth s equivalence problem in unit regular rings, Proc. Amer. Math. Soc ) [11] R. Hartwig, Roth s removal rule revisited, Linear Algebra Appl ) [12] D. Huylebrouck, The generalized inverses of a sum with Radical elements: applications, Linear Algebra Appl ) [13] R.M. Guralnick, Matrix equivalence and isomorphism of modules, Linear Algebra Appl ) [14] R.M. Guralnick, Roth s theorems and decomposition of modules, Linear Algebra Appl ) [15] R.M. Guralnick, Roth s theorems for sets of matrices, Linear Algebra Appl ) [16] W. Gustafson, J. Zelmanowitz, On matrix equivalence and matrix equations, Linear Algebra Appl ) [17] W. Gustafson, Roth s theorems over commutative rings, Linear Algebra Appl ) [18] H.K. Wimmer, Roth s theorems for matrix equations with symmetry constraints, Linear Algebra Appl ) [19] H.K. Wimmer, Linear matrix equations: the module theoretic approach, Linear Algebra Appl ) [20] H.K. Wimmer, Consistency of a pair of generalized Sylvester equations, IEEE Trans. Automat. Control ) [21] M.A. Beitia, J.M. Gracia, Sylvester matrix equation for matrix quadruples, Linear Algebra Appl ) [22] L. Huang, J. Liu, The extension of Roth s theorem for matrix equations over a ring, Linear Algebra Appl ) [23] Q.W. Wang, Roth s theorems for centroselfconjugate and centroskewselfconjugate solutions to systems of linear matrix equations over a finite dimensional central algebra, Southeast Asian Bull. Math. 27, in press. [24] Q.W. Wang, J.H. Sun, S.Z. Li, Consistency for biskew) symmetric solutions to systems of generalized Sylvester equations over a finite central algebra, Linear Algebra Appl ) [25] Q.W. Wang, On the center skew-)self-conjugate solutions to the systems of matrix equations over a finite dimensional central algebra, Math. Sci. Res. Hot-Line 5 12) 2001) [26] Q.W. Wang, S.Z. Li, Persymmetric and perskewsymmetric solutions to sets of matrix equations over a finite central algebra, Acta Math. Sinica 47 1) 2004)

12 54 Q.-W. Wang / Linear Algebra and its Applications ) [27] Q.W. Wang, J.H. Sun, The consistency of systems of matrix equations over a finite central algebra, J. Natur. Sci. Math. 41 1) 2001) [28] J.K. Baksalary, R. Kala, The matrix equation AXB CYD = E, Linear Algebra Appl ) [29] A.B. Özgüler, The matrix equation AXB CYD = E over a principal ideal domain, SIAM J. Matrix Anal. Appl ) [30] L. Huang, Q. Zeng, The solvability of matrix equation AXB CYD = E over a simple Artinian ring, Linear and Multilinear Algebra ) [31] L. Huang, The solvability of matrix equation AXB CYD = E over a ring, Adv. Math. China) 26 3) 1997) in Chinese). [32] B. Brown, N.H. McCoy, The maximal regular ideal of a ring, Proc. Amer. Math. Soc )