Common Solutions of a Pair of Matrix Equations
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1 Applied Mathematics E-Notes, 2(22), c ISSN Available fee at mio sites of amen/ Common Solutions of a Pai of Matix Equations Yong-ge Tian Received 21 Octobe 21 Abstact Solvability condition and common solution of the pai of linea matix equations AX + X = M and AX = C ae detemined by making use of anks and genealized inveses of matices. Some of thei applications to genealized inveses of matices ae also pesented. 1 Intoduction We conside in this aticle common solutions of the pai of simultaneous matix equations AX + X = M, (1) AX = C, and pesent some of thei applications to genealized inveses of matices. The fist equation in (1) is called the Sylveste equation in the liteatue and is widely studied, see [3] and the efeences theein fo its histoy and applications. The second equation in (1) is also well known in the liteatue, see [1, 2, 9]. A diect motivation fo us to conside the common solutions of the pai of matix equations in (1) aises fom chaacteizing vaious commutativity fo genealized inveses of matices, such as, AA = A A, A k A = A A k, and A D A = A A D, AA = A A and so on, as well as factoizations of matix with the fom M = AA A A o M = A k A A A k and so on. Note that genealized invese (inne invese) A is a solution to the matix equation AXA = A. Hence the equalities mentioned above can be egaded as special cases of (1). Thoughout, C denotes the field of complex numbes. R(A), (A), A and A as usual denote the ange (column space), the ank, the conjugate tanspose, and a genealized invese of matix A, espectively. Moeove, we denote E A = I AA and F A = I A A fo any A. The following ank fomulas ae due to Masaglia and Styan [6, Theoem 19]. LEMMA 1.1. Let A C m n, C m k and C C l n be given. Then (a) [ A, ]=(A)+(E A )=()+(E A). A (b) = (A)+(CF C A )=(C)+(AF C ). Mathematics Subject Classifications: 15A9, 15A24, 15A27. Depatment of Mathematics and Statistics, Queen s Univesity, Kingston, Ontaio, Canada K7L 3N6 147
2 148 Matix Equations A (c) = ()+(C)+(E C AF C ). Fom Lemma 1.1(c), we obtain A E C1 C F 1 = A C C 1 ( 1 ) (C 1 ). (2) 1 Lemma 1.1 and (2) ae quite useful in simplifying vaious ank equalities involving genealized inveses of matices. The following esult on the matix equation AX = C is also well known, see [1, 2, 9]. LEMMA 1.2. The following five statements ae equivalent: (a) The matix equation AX = C is consistent. (b) AA C = C and C = C. (c) AA C = C. (d) R(C) R(A) andr(c ) R( ). (e) [ A, C ]=(A) and = (). C In case one of the five statements in Lemma 1.2 holds, the geneal solution of AX = C canbeexpessedinthefomx = A C + V A AV, o X = A C + F A V 1 + V 2 E, whee V, V 1 and V 2 ae abitay matices. LEMMA 1.3. Let A C m p, C q n,c C m,d C s n and N C m n be given. Then (a) The matix equation AX + CY D = N (3) is solvable if and only if the following fou ank equalities hold [8] [ A, C, N ]=[ A, C ], D =, (4) D N N A D N C = (A)+(D), = ()+(C). (5) (b) In case (3) is solvable, the geneal solutionofeq.(3)canbeexpessedinthe fom [11, 12] X = X + X 1 X 2 + X 3 and Y = Y + Y 1 Y 2 + Y 3, (6) whee X and Y ae two special solutions of Eq.(3), X 1,X 2,X 3 and Y 1,Y 2,Y 3 ae the geneal solutions of the following fou homogeneous matix equations AX 1 CY 1 =, X 2 + Y 2 D =, AX 3 =, CY 3 D =, (7)
3 Y. G. Tian 149 o explicitly X = X +[I p, ]F G UE H Iq + F A V 1 + V 2 E, (8) Y = Y +[,I ]F G UE H + F I C W 1 + W 2 E D, (9) s whee X and Y ae two special solutions of Eq.(3), G =[A, C ], H=,U, V D 1, V 2,W 1 and W 2 ae abitay. Some expessions of special solutions of (3) wee given in [1] and [8]. ut we only need (8) and (9). 2 Main Results Ou fistmainesultisasfollows. THEOREM 2.1. Let A C m m, C n n and C, M C m n be given. Then (a) The matix equations AX + X = M, AX = C (1) have a common solution X if and only if A,, C, M satisfy the following six conditions R(C) R(A), R(C ) R( M A ), = (A)+(), (11) AC + C = AM, R( C AM ) R(A 2 ), R[( C M) ] R[( 2 ) ]. (12) (b) In case (11) and (12) hold, the geneal common solution of (1) can be expessed in the fom In X = X +[F A, ]F G UE H +[,I m ]F G UE H + F E A SE, (13) whee X is a special solution of Eq.(1), G=[F A, A ], H =,Uand S ae E abitay. (c) The equations in (1) have a unique common solution if and only if A and ae nonsingula and AC + C = AM. In this case, the unique common solution is X = A 1 C 1. PROOF. Suppose fist that (1) has a common solution. This implies that AX + Y = C and AX = C ae solvable espectively. Thus (11) follows diectly fom Lemmas 1.2 and 1.3. Pe- and post-multiplying A and of the both sides of AX + X = M, espectively, yield A 2 X = AM C and X 2 = M C, which imply the two ange inclusions in (12). Consequently, pe- and post-multiplying A and on the both sides of AX + X = M poduces the fist equality in (12).
4 15 Matix Equations We next show that unde (11) and (12), the two equations in (1) has a common solution and thei geneal common solution can be witten as (13). y Lemma 1.2, the geneal solution of AX = C unde (11) is X = A C + F A V 1 + V 2 E, (14) whee V 1 and V 2 ae abitay. Substituting it into AX + X = M yields AV 2 E + F A V 1 = M C A C. (15) y Lemma 1.3, this equation is solvable if and only if it satisfies the following fou ank equalities [ A, F A,N]=[ A, F A ], E =, (16) E N and N A N FA = (A)+(), = (F E A )+(E ), (17) whee N = M C A C. Simplifying them by Lemma 1.1 and (2), we find that A Im M A [ A, F A,N]= C A E N A Im [ A, F A ]= A = N E A (A) =[ A 2,C AM ]+m (A), (A) =(A 2 )+m (A), I n () = M C = I n M C = A C A 2 C M () =( 2 )+n (), M = + n (), A, N FA E = M C A C I m I n (A) () A = I m I n (A) () AM AC C = m + n + (AM AC C) (A) (), and (F A )+(E )=m + n (A) ().
5 Y. G. Tian 151 Substituting them into (16) and (17) yields the esults in (11) and (12). This fact implies that unde (11) and (12), the equation (15) is solvable. Solving fo V 1 and V 2 in (15) by Lemma 1.3, we obtain thei geneal solutions V 1 = V 1 + A AS 1 + S 2 E +[ I m, ](I [ F A, A ] [ F A, A ])U I E E In, V 2 = V 2 + F A T 1 + T 2 +[,I m ]( I [ F A, A ] [ F A, A ])U I E E I n, whee V 1 and V 2 ae two special solutions of (15), U, S 1,S 2,T 1 and T 2 ae abitay. Substituting them into (14) yields X = A C In + F A V 1 + V 2 E +[F A, ]F G UE H +F A S 2 E +[,I m ]F G UE H + F E A T 1 E, which can also be witten in the fom of (13). The poof is complete. Some diect consequences can be deived fom the above theoem. Hee ae some of them. COROLLARY 2.2. Let A, M C m m be given. Then (a) Thee is A such that if and only if A and M satisfy the following fou conditions M = AA A A (18) AMA =, R(A + AM) R(A 2 ), R[(A MA) ] R[(A 2 ) M A ], =2(A). A (b) Unde (A 2 )=(A), thee is A such that Eq.(18) holds if and only if M A AMA =and =2(A). A (c) Thee is A such that if and only if A and M satisfy the following fou conditions M = AA + A A (19) 2A 2 = AMA, R( A AM ) R(A 2 ),
6 152 Matix Equations R[( A MA) ] R[(A 2 ) M A ], =2(A). A (d) Unde (A 2 )=(A), thee is A such that Eq.(19) holds if and only if 2A 2 M A = AMA and =2(A). A (e) Thee is A such that AA = A A holds if and only if (A 2 )=(A) [13]. Indeed, applying Theoem 2.1 to the system AX XA = M and AXA = A yields the esults in the coollay. An extension of Theoem 2.1 is given below. Its poof is simila to that of Theoem 2.1 and is theefoe omitted. THEOREM 2.3. Let A C m k, C l n,a 1 C k k, 1 C l l,c C m n,m C k l and suppose that Then the following matix equations R(A 1) R(A )andr( 1 ) R(). (2) A 1 X + X 1 = M, AX = C (21) have a common solution X if and only if the following conditions ae satisfied: M A1 1 R(C) R(A), R(C ) R( ), (22) = (A 1 )+( 1 ),AA 1 A C + C 1 = AM, (23) R( AM C 1 ) R(AA 1 ), R[( M A 1 A C ) ] R[( 1 ) ]. (24) COROLLARY 2.4. Let A, M C m m be given. Then thee is A such that M = A k A A A k (25) if and only if A and M satisfy the following fou conditions AMA =, R( A k + AM ) R(A k+1 ), R[(A k MA) ] R[(A k+1 ) ], (26) M A k A k =2(A k ). (27) In paticula, thee is A such that A k A = A A k holds if and only if (A k+1 )=(A k ) [13]. PROOF. In fact, (25) is equivalent to A k X XA k = M and AXA = A. Thus (25) (27) follow fom (2) (24). COROLLARY 2.5. Let A C m m be given. Then thee exists A such that A D A = A A D, whee A D is the Dazin invese of A.
7 Y. G. Tian 153 PROOF. It is obvious that A D A = A A D is equivalent to A D X = XA D and AXA = A. (28) Note that R(A D ) R(A) andr[(a D ) ] R(A ). Then applying Theoem 2.3 to (28) yields the desied esult. COROLLARY 2.6. Let A, C m m be given. Then thee is A such that AA = A A (29) if and only if (AA) =(A) =(A). (3) PROOF. The equality (29) is equivalent to the pai of matix equations AX = XA and AXA = A. Thus (3) is deived fom Theoem 2.3. In paticula when istakensuchthat(aa) =(A), thee exists A satisfying (29). In this case, this genealized invese is called the commutative genealized invese of A with espect to and is denoted by A, which was examined in Khati [5]. Note that (AA A) = (AA ) = (A A) = (A). Thus any squae matix A has a commutative genealized invese A A. In fact, the Mooe-Penose invese A is a special case of the commutative genealized invese A A. Acknowledgments. The autho would like to thank the efeee fo valuable suggestions. The eseach of the autho was suppoted in pat by the Natual Sciences and Engineeing Reseach Council of Canada. Refeences [1] J. K. aksalay and R. Kala, The matix equation AX + CY D = E, Linea Algeba Appl., 3(198), [2] A. en-isael and T. N. E. Geville, Genealized Inveses: Theoy and Applications, R. E. Kiege Publishing Company, New Yok, 198. [3] R. hatia and P. Rosenthal, How and why to solve the opeato equation AX X = Y, ull London Math. Soc., 29(1997), [4] C. G. Khati, A note on a commutative g-invese of matix, Sankhyā Se. A, 32(197), [5] C. G. Khati, Commutative g-invese of a matix, Math. Today, 3(1985), [6] G. Masaglia and G. P. H. Styan, Equalities and inequalities fo anks of matices, Linea and Multilinea Algeba, 2(1974),
8 154 Matix Equations [7] S. K. Mita, A pai of simultaneous linea matix equations A 1 X 1 = C 1 and A 2 X 2 = C 2 and a pogamming poblem, Linea Algeba Appl., 131(199), [8] A.. Özgüle, The equations AX + CY D = E ove a pincipal ideal domain, SIAM J. Matix Anal. Appl., 12(1991), [9] C.R.Rao and S.K.Mita,Genealized Invese of Matices and Its Applications, Wiley, New Yok, [1] R. E. Roth, The equations AX Y= C and AX X = C in matices, Poc. Ame. Math. Soc., 3(1952), [11] Y. Tian, The geneal solution of the matix equation AX = CY D, Math. Pactice and Theoy, 1(1988), [12] Y. Tian, Solvability of two linea matix equations, Linea and Multilinea Algeba, 48(2), [13] Y. Tian, The maximal and minimal anks of some expessions of genealized inveses of matices, Southeast Asian ull. Math., 25(22),
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