Intenational eseach Jounal of Pue Algeba -() 01 98-0 Available online though wwwjpainfo ISSN 8 907 SOE ESULTS ON THE GOUP INVESE OF BLOCK ATIX OVE IGHT OE DOAINS Hanyu Zhang* Goup of athematical Jidong second middle school Jidong 15800 P China (eceived on: 9-01-1; evised & Accepted on: 07-0-1) Suppose be a ight Oe domain with identity 1 and ABSTACT denote the set of all pape we give the existences and the epesentations of the goup invese fo block matix BA C matices ove In this AC C and unde the special condition ove ight Oe domains The pape's esults genealize some elative esults of Wang and Fan (Intenational eseach Jounal of Pue Algeba :7-51 01) Keywod: goup invese; block matix; ight Oe domain 1 INTODUCTION A squae matix G is said to be goup invese of A if G satisfies AGA = A GAG = G and AG = GA It is well known that if G exists it is unique We then wite G = A When A exists we denote A π = I AA The Dazin inveses and goup inveses of block matices have applications in many aeas especially in singula diffeential and diffeence equations and finite akov chains (see [-8]) It is impotant to study them in a lage ing In 001 Cao [11] studied the poblem ove a division ing Zhang and Bu [1] made a eseach ove a ight Oe domain in 01 The pupose of this pape extends the esults of goup invese ove skew fields given in [10] to ight Oe domains A ing is called a ight Oe domain if it possesses no zeo divisos and evey two elements of the ing have a ight common multiple A left Oe domain is defined similaly Evey ight(left) Oe domain can be embedded in the skew field(denoted by K ) of quotients of itself Fo any matix A ove the ank of A (denoted by ( A )) is defined as the ank of A ove K (see [1]-[]) Let be a ight Oe domain with identity 1 be the set of all m n matices ove The ank of a matix A (denoted by ( A )) is defined as the ank of A ove K ie the maximum ode of all invetible subblocks of A ove K A matix A is called egula if thee exists a matix X such that AXA = A then X is called a {1}- invese o egula invese of A In this case denote the set of all {1}-inveses of A by A {1} Let A be any {1}- invese of A Let A denote the ange and the ow ange of a matix A by A ( ) and ( A ) n 1 ( A) { Ax x 1 = } and ( A) = { ya y m } SOE LEAS In this section we give some lemmas which play impotant ole thoughout this pape *Coesponding autho: Hanyu Zhang* Goup of athematical Jidong second middle school Jidong 15800 P China E-mail: zhanghanyu@16com Intenational eseach Jounal of Pue Algeba-Vol-() Feb 01 98
Hanyu Zhang* / Some esults On The Goup Invese Of Block atix Ove ight Oe Domains/IJPA- () Feb-01 [9] Lemmas: 1 Let (i) A exists; (ii) AX= A fo some (iii) YA = A fo some (iv) A ( ) = A ( ); (v) ( A) = ( A ) Lemmas: Let ( BA ) exist A the followings ae equivalent: X Y In this case In this case A A = AX ; Y A = ; If AB and BA ae all egula ( A) ( B) ( AB) ( BA) AB = = = then Poof: If ( A) = ( B) = ( AB) = ( BA) then thee exist matices XYZ and W ove A = ABX = YBA and B = BAZ = WAB Thus AB( AB) A = ABX = A BA( BA) B = BAZ = B B( AB) AB = WAB = B Thus A ( ) AB ( ) ( A) = ( BA) ( B) = ( AB) Theefoe ( AB) = A( B) = A( BA) = AB( A) = AB( AB) = ( ABAB) ( BA) = ( B) A = ( AB) A = ( A) BA = ( BA) BA = ( BABA) By Lemma 1 we conclude that n m A Lemmas: Let (i) ( AB) = A[( BA) ] B ; (ii) ( AB) A = A( BA) ; (iii) B( AB) A = BA( BA) ; (iv) (v) (vi) AB( AB) A = A; A( BA) BA = A ; BA( BA) B = B B ( BA ) both exist Poof: Fom Lemma of [9] they ae obvious AIN ESULTS Theoem: 1 Let ( BA ) exist then AC n n = and B ( ) C ( ) then C exists if and only if B ( ) = C ( ) = BC ( ) = CB ( ) and BC CB ae both egula 1 exists then = π 1 = ( BC) AC( BC) π = ( BC) B ( BC) [ AC( BC) ] B = C( BC) = C( BC) AC( BC) B () If Poof: : Only if pat If has a goup invese by Lemma 1 thee exist matices X and Y ove such that I A BC 0 = 0 I CAC CB K such that A( BA) BA = YBA = A = B ( ) = BA ( ) = X = Y Since 01 JPA All ights eseved 99
Let Then ie Hanyu Zhang* / Some esults On The Goup Invese Of Block atix Ove ight Oe Domains/IJPA- () Feb-01 I A 0 B = 0 I C 0 X1 X X = X X I A Y1 Y I A Y = 0 I Y Y 0 I BC 0 X X 0 = Y1 Y BC 0 0 Y Y = CAC CB C 1 CAC CB X X C BCX 1 = 0 BCX B = () CACX1+ CBX = C () CACX CBX 0 + = () Y1BC + Y CAC = B (5) Y CB B = (6) YBC + Y CAC = C (7) YCB = 0 (8) exists it is easy to know B is egula Using () and (6) we have Note that and CB = CBB B = CBB Y CB ie both BC and CB ae egula BC = BB BC = BCX B BC Fom () and (6) we know B ( ) = BC ( ) ( B) = ( CB) Theefoe B ( ) = BC ( ) = CB ( ) and C ( ) BC ( ) = B ( ) C ( ) ie B ( ) = C ( ) The if pat By Lemma we know that ( BA) both exist Let X1 ( BC) π = B X = ( BC) B X = ( CB) C X = ( CB) CAC( BC) B That implies = X have a solution so by Lemma 1 exists can be got fom X AC 0 ( BC) = X C ( CB) C ( CB) CAC( BC) B B( CB) C AC( BC) B B( CB) CAC( BC) 0 ( BC) = 0 C( BC) B ( CB) C ( CB) CAC( BC) B (): By Lemma 1 the expession of = Using Lemma next we can compute that 01 JPA All ights eseved 00
Hanyu Zhang* / Some esults On The Goup Invese Of Block atix Ove ight Oe Domains/IJPA- () Feb-01 1 = Coollay: 1 Let () If A B = A n n and B ( ) A ( ) exists if and only if ( B) ( A) ( BA) ( AB) exists then π 1 = ( BA) A( BA) π = ( BA) B ( BA) [ A( BA) ] B = A( BA) = A( BA) A( BA) B then = = = and BA AB ae both egula 1 = Poof: The esults is a special case of Theoem 1 let A= I in Theoem the conclusion is obvious Similaly we can get the following esults BA n n = and B ( ) C ( ) then C exists if and only if B ( ) = C ( ) = BC ( ) = CB ( ) and BC CB ae both egula 1 exists then = Theoem: Let () If 1 = B( CB) A B( CB) AB( CB) B = B( CB) = CB( CB) AB( CB) A + ( CB) B CB[( CB) AB] ( CB) B = CB( CB) AB( CB) A A n n = and B ( ) A ( ) then B exists if and only if ( B) = ( A) = ( BA) = ( AB) and BA AB ae both egula 1 exists then = Coollay: Let () If 1 = A( BA) A( BA) A( BA) A = A( BA) = BA( BA) A( BA) + ( BA) A BA[( BA) A] ( BA) A = BA( BA) A( BA) Poof: Let A= I in Theoem 01 JPA All ights eseved 01
Hanyu Zhang* / Some esults On The Goup Invese Of Block atix Ove ight Oe Domains/IJPA- () Feb-01 EFEENCES [1] K Zhang C Bu Goup inveses of matices ove ight Oe domains Appl ath Comput 01 18: 69-695 [] P Cohn Fee ings and Thei elations London athematical Society onogaphs second ed vol 9 Academic Pess Inc London 1985 [] L Huang Geomety of atices Ove ing Science Pess Beijing 006 [] S L Campbell CD eye Genealized Inveses of Linea Tansfomations Dove Newyok 1991 [5] A Ben-Isael TNE Geville Genealized Inveses: Theoy and Applications second ed Spinge-Velag Newyok 00 [6] SL Campbell The Dazin invese and systems of second ode linea diffeential equations Linea ultilinea Algeba 198 1: 195-198 [7] C D eye The ole of the goup genealized invese in the theoy of finite akov chains SIA ev 1975 17(): -6 [8] C Bu K Zhang epesentations of the Dazin invese on solution of a class singula diffeential equations Linea ultilinea Algeba 011 59: 86-877 [9] Y Ge H Zhang Y Sheng C Cao Goup invese fo two classes of anti-tiangula block matices ove ight Oe domains J Appl ath Comput 01 : 18-191 [10] J Wang X Fan Some esults on the goup invese of block matix ove skew fields Intenational eseach Jounal of Pue Algeba 01 : 7-51 [11] C Cao Some esults of goup inveses fo patitioned matices ove skew fields Sci Heilongjiang Univ 001 18: 5-7 (in chinese) Souce of Suppot: Nil Conflict of inteest: None Declaed 01 JPA All ights eseved 0