Filomat 28:6 2014, 1103 1112 DOI 102298/FIL1406103D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat The Expression for the Group Inverse of the Anti triangular Block Matrix Fapeng Du a, Yifeng Xue b a School of mathematical & Physical Sciences, Xuzhou Institute of Technology, Xuzhou, 221008, Jiangsu Province, P R hina b Department of Mathematics, East hina Normal University, Shanghai, 200241,P R hina Abstract In this paper, we present the explicit expression for the group inverse of the sum of two matrices As an application, the explicit expression for the group inverse of the anti-triangular block matrix D 0 B and are obtained without any conditions on sub-blocks D 1 Introduction Let M n be the set of all n n matrix on complex field and let I denote the unit of M n For an element A M n, if there is an element B M n which satisfies ABA A, then B is called a {1} inverse of A If ABA A and BAB B hold, then B is called a {2} inverse of A, denoted by A + An element B is called the Drazin inverse of A, if B satisfies A k BA A k ; BAB B; AB BA for some integer k B is denoted by A D The least such integer k is called the index of A, denoted by inda We denote by A π I AA D the spectral idempotent of A In the case inda 1, A D reduces to the group inverse of A, denoted by A # The Drazin inverse has various applications in singular differential equations and singular difference equations, Markov chains, and iterative methods see[4 7, 9 11, 15, 16, 20] In 1979, S ampbell and Meyer proposed an open problem to find an explicit representation for the Drazin inverse of a 2 2 block A matrix in terms of its sub-blocks, where A and D are supposed to be square matricessee [4] A B D A simplified problem to find an explicit representation for the Drazin inverse of was proposed by S B 0 ampbell in 1983see [7] Until now, both problems have not been solved However, many authors have considered the two problems under certain conditions on the sub-blockssee [3, 9, 12, 13, 15, 17, 18, 25] As 2010 Mathematics Subject lassification Primary 15A09; Secondary 65F20 Keywords block matrix, generalized inverse, group inverse Received: 11 October 2013; Accepted: 25 March 2014 ommunicated by Dragana vetkovic Ilic Research supported by Jiangsu Province University NSF 14KJB110025 Email addresses: jsdfp@163com Fapeng Du, yfxue@mathecnueducn Yifeng Xue
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1104 a special case, the expression for the group inverse of 2 2 block matrix also has been studied under some conditionssee [1, 2, 8, 14, 18, 19, 23] In this paper, we give the explicit expression for the group inverse of the sum of two matrices As 0 B an application, the expression for the group inverse of the anti-triangular matrix and are D presented without any conditions on sub-blocks 2 Preliminaries In this section, we present some important lemmas and investigate the expression of the group inverse in term of its generalized inverse Let us begin with a familiar lemma Lemma 21 [19] Let A, B M n, then I+AB is invertible iff I+BA is invertible and I+AB 1 I AI+BA 1 B Lemma 22 Let P be an idempotent matrix in M n, then I P is invertible iff P 0 Proof Since I P is invertible, there is an X M n such that XI P I, that is, X XP I 0 XP XP P So, Lemma 23 [24, Theorem 459] Let A M n \{0} Then the following conditions are equivalent: 1 A # exists 2 AA + + A + A I is invertible for some A + 3 A 2 A + + I AA + is invertible for some A + 4 A 2 A + + I AA + is invertible for any A + 5 A + I AA + is invertible for some A + 6 A + I AA + is invertible for any A + 7 A + A 2 + I A + A is invertible for some A + 8 A + A 2 + I A + A is invertible for any A + 9 A + I A + Ais invertible for some A + 10 A + I A + Ais invertible for any A + Proof The equivalence of 1 to 4 are presented in [24, Theorem 459] Noting that A 2 A + + I AA + I + A IAA + and A + I AA + I + AI A +, by Lemma 21, we have 3 5, 4 6, 7 9, 8 10, 5 9, 6 10 Lemma 24 Let A M n \{0} If A # exists then A # AI + A A + A 2 I + A AA + 2 A I + A AA + 1 AI + A A + A 1, independent of the choice of A +
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1105 Proof Put W A 2 A + + I AA + By Lemma 23, W is invertible Set B W 2 A Noting that AA + W WAA + A 2 A +, WA A 2, we have AB AW 2 A AW 2 AA + A A 2 A + W 2 A AA + WW 2 A W 1 A, BA W 2 A 2 W 2 WA W 1 A, ABA W 1 A 2 W 1 WA A, BAB W 1 AW 2 A W 1 AAA + W 2 A W 1 AA + W 1 A W 2 A The above indicate that B A # and independent of the choice of A + Noting that I + A AA + 1 A AI + A A + A 1, by Lemma 21, we have A # A 2 A + + I AA + 2 A [I + A IAA + ] 2 A [I A II + A AA + 1 AA + ] 2 A [I A II + A AA + 1 AA + ]2I AA + I + A AA + 1 A [I A II + A AA + 1 AA + ]AI + A A + A 1 AI + A A + A 2 I + A AA + 2 A I + A AA + 1 AI + A A + A 1 Proposition 25 Let A, B M n Then AB # exists iff I + AB ABB + A + ABB + + A + is invertible iff I + AB B + A + ABB + + A + AB is invertible In this case, AB # {I + AB ABB + A + ABB + + A + } 2 AB AB{I + AB B + A + ABB + + A + AB} 2, Proof Noting that B + A + ABB + + A + is a {1,2} inverse of AB By Lemma 24, we get the results orollary 26 Let A, B M n with AB # exists 1 If B is invertible, then AB # exists iff I + AB AA + is invertible In this case, AB # I + AB AA + 2 AB 2 If A is invertible, then AB # exists iff I + AB B + Bis invertible In this case, AB # ABI + AB B + B 2 3 Main Results Let A, B,, D M n Throughout of this paper, we denote E A I AA +, F A I A + A Lemma 31 Let A, X, Y M n and Z I YA + X, U E A X, V YF A, S E V ZF U Let G A + F A V + YA + + F A V + Z + A + X[F U S + E V YA + I F U S + E V ZU + E A ] Then G is a {1} inverse of A XY
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1106 Proof A XYG AA + + AA + X[F U S + E V YA + I F U S + E V ZU + E A ] XE V YA + XVV + Z + XYA + X[F U S + E V YA + I F U S + E V ZU + E A ] AA + + X U[F U S + E V YA + I F U S + E V ZU + E A ] XE V YA + XVV + Z + XYA + X[F U S + E V YA + I F U S + E V ZU + E A ] AA + + X[F U S + E V YA + I F U S + E V ZU + E A ] + UU + E A XE V YA + XVV + Z + XI Z[F U S + E V YA + I F U S + E V ZU + E A ] AA + + X[F U S + E V YA + I F U S + E V ZU + E A ] + UU + E A XE V YA + X XE V Z[F U S + E V YA + I F U S + E V ZU + E A ] AA + + UU + E A XE S E V YA + XE V ZU + E A + XSS + E V ZU + E A AA + + UU + E A XE S E V YA + XE S E V ZU + E A AA + + UU + E A XE S E V YA + + ZU + E A I E U E A XE S E V YA + + ZU + E A GA XY A + A F A V + YA + A + F A V + Z + A + XF U S + E V YA + A A + XY + F A V + YA + XY F A V + Z + A + X[F U S + E V YA + I F U S + E V ZU + E A ]XY A + A F A V + Y + F A V + V + F A V + Z + A + XF U S + E V Y A + XY + F A V + YA + XY F A V + Z + A + X[F U S + E V YA + I F U S + E V ZU + E A ]XY A + A F A V + Y + F A V + V + F A V + Z + A + XF U S + E V Y A + XY + F A V + I ZY F A V + Z + A + X[F U S + E V I ZY I F U S + E V ZU + UY] A + A + F A V + V A + X + F A V + ZY + F A V + Z + A + X[F U S + E V ZY + I F U S + E V ZU + UY] A + A + F A V + V + F A V + Z + A + X[F U S + E V ZY + I F U S + E V ZU + UY Y] A + A + F A V + V + F A V + Z + A + X[F U S + E V Z F U S + E V ZU + U F U ]Y A + A + F A V + V F A V + Z + A + XF U F S Y I F A F V F A V + Z + A + XF U F S Y It is easy to verify A XYGA XY A XY This shows G is a {1} inverse of A XY orollary 32 Let A, X M n and Z I + A + X, U E A X, S A + AZF U Let and G A + + F A A + X[F U S + A + + I F U S + A + AZU + E A ], A + XG I E U E A + XE S A + A + AZU + E A, GA + X I F U F S Proof Replacing Y by I in Lemma 31, we get easily the first part of the results Noting that Z I + A + X and S A + AZF U, we have GA + X I F A A + XF U F S I I A + A A + XF U F S I [I A + AI + A + X]F U F S I I A + AZF U F S I F U F S
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1107 Similarly, we have orollary 33 Let A, X M n and Z I + XA +, V XF A, S E V ZAA + Let G A + F A V + XA + + F A V + Z + A + X[AA + S + E V XA + + I AA + S + E V ZE A ] Then G is a {1} inverse of A + X and A + XG I E S E V, GA + X I F A F V F A V + ZAA + A + F S X Theorem 34 Let A, X, Y M n and Z I YA + X, U E A X, V YF A, S E V ZF U Then A XY # exists iff A XY + E U E A + XE S E V YA + + ZU + E A is invertible iff is invertible and A XY + F A F V + F A V + Z + A + XFuF S Y A XY # {A XY + E U E A + XE S E V YA + + ZU + E A } 2 A XY A XY{A XY + F A F V + F A V + Z + A + XFuF S Y} 2 {A XY + E U E A + XE S E V YA + + ZU + E A } 1 A XY {A XY + F A F V + F A V + Z + A + XF U F S Y} 1 Proof By Lemma 31, G is a {1} inverse of A XY Thus, A XY + GA XYG is a generalized inverse of A XY Hence, A XYA XY + A XYG I E U E A XE S E V YA + + ZU + E A, A XY + A XY GA XY I F A F V F A V + Z + A + XF U F S Y So, the results follow by Lemma 23 and Lemma 24 Using Lemma 23, Lemma 24 and orollary 32, orollary 33, we have the following corollaries: orollary 35 Let A, X M n and Z I + A + X, U E A X, S A + AZF U Then A + X # exists iff is invertible iff A + X + F U F S is invertible and A + X # A + XA + X + F U F S 2 A + X + E U E A XE S A + A + AZU + E A {A + X + E U E A XE S A + A + AZU + E A } 2 A + X {A + X + E U E A XE S A + A + AZU + E A } 1 A + XA + X + F U F S 1 orollary 36 Let A, X M n and Z I + XA +, V XF A, S E V ZAA + Then A + X # exists iff is invertible iff A + X + E S E V is invertible and A + X # A + X + E S E V 2 A + X A + X + F A F V + F A V + ZAA + A + F S X A + X{A + X + F A F V + F A V + ZAA + A + F S X} 2 A + X + E S E V 1 A + X{A + X + F A F V + F A V + ZAA + A + FsX} 1
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1108 In the following, we investigate the expression for the group inverse of anti-triangular matrix First, we cite a lemma which comes from [21] Lemma 37 [21] Suppose M, X M n Then N M MXM has a {1} inverse Y iff M has a {1} inverse X + I XMYI MX Lemma 38 Let M and Z D A D + B, P E, Q F A, R ZF P, W E R QQ + Then there exists a {1} inverse G of M such that E I MG P E A 0 E W E R ZP + E A + A + E W E R A + 0 Proof Taking X and N M MXM Let Y be a {1} inverse of N Then by Lemma 37, G X + I XMYI MX be a {1} inverse of M and I MG I MX MI XMYI MX I NYI MX 1 0 P 0 E Let N 1, N Q 0 2 Then N N 0 Z 1 + N 2 F A D A + B Note that 0 Q N + 1 + I 0 P +, T I + N 0 2 N + 1 ZP +, I PP N 1 N + 1 + 0 Q 0 QQ +, N + 1 N 1 + Q 0 0 P +, P PP V N 2 F N1, S E 0 ZF V TN 1 N + 1 + 0 P E R ZP + E R QQ + and S + PP + 0 W + E R ZP + W + Thus, by orollary 33, we have Hence, by Eq1, we have I MG I NYI MX E P 0 EA 0 E W E R ZP + E W E R A + I E P E A 0 E W E R ZP + E A + A + E W E R E I NY E S E V P 0 E W E R ZP + E W E R A 0 Theorem 39 Let M with A D #, D # exist Then M # exists iff D π A π 0 In this case, M # A # 0 D # 2 A π + D π A # 2 D # A # D #
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1109 Proof Take A + A #, D + D # and B 0 in Lemma 38, we have Z D, P 0, Q A π, R Z D, W D π QQ + and A + A M + I MG π 0 E W D π A + D + E W D π Since A+A π and D+D π are invertible, M+I MG is invertible iff I WW + D π is invertible Thus, by Lemma 22, I WW + D π is invertible iff WW + D π 0 iff W 0 iff D π Q D π A π 0 Hence, if M # exists, then A M + I MG 1 # + A π 0 D π A + D # + D π By simple calculation, we get M # A # 0 D # 2 A π + D π A # 2 D # A # D # 0 B Theorem 310 Let M and R DF D B, W E R + Then M # exists iff DF B B + E W E R F B + DB + B is invertible In this case, Here, 2 I + Bξη Bξ 0 B M # B + F B ξη F B ξ D ξ {DF B B + E W E R F B + DB + B} 1, η + DB + + E W E R I DB + Proof Take A 0 in Lemma 38, we have Z D, P B, Q, R DF B, W E R + and E M + I MG B B E W E R DB + D + E W E R I B I 0 + DB + + E W E R I DB + D + E W E R B + I I 0 EB B + DB + + E W E R I DB + DF B B + E W E R F B + DB + B B + I By Lemma 23, we have M # exists iff DF B B + E W E R F B + DB + B is invertible Put ξ {DF B B + E W E R F B + DB + B} 1, η + DB + + E W E R I DB + Then Thus, by Lemma 31, we have I + Bξη Bξ M + I MG 1 B + F B ξη F B ξ 2 I + Bξη Bξ 0 B M # B + F B ξη F B ξ D
We find it is very difficulty to calculate as following: Theorem 311 Let M FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1110 # if take D 0 in Lemma 38 So, we present another method and P AF, W E P BB + Then M # exists iff AF B + E W E P F + A + is invertible In this case, Here, M # F ξ + 2 F ξη ξ I + ξη ξ {AF B + E W E P F + A + } 1, η B + A + + E W E P I A + 0 B A 0 Proof Let M 1, M 2 Then M M 1 + M 2 Let I 2 be the identity of M 2 and P AF, W E P BB + Noting that 0 B P 0 M + 1, V M 2 F M1, E V I 2 VV + EP 0 I A, Z I 2 + M 2 M + 1 +, W S E V ZM 1 M + 1 EP A + W 0 +, S + + W + E P A + 0 +, EW E E S W E P A + A + EW E, M + E 0 E S E V P B E W E P A + E and A + EW E M + E S E V P B E W E P A + E A + EW E P B E W E P A + + A + + E W E P + I + I AF B + E W E P F + A + B + A + + E W E P I A + I + I AF B + E W E P F + A + B + A + + E W E P I A + I + E Thus, by orollary 36, we have M # exists iff AF B + E W E P F + A + is invertible Put ξ {AF B + E W E P F + A + } 1, η B + A + + E W E P I A + Then, by Simple calculation, we have M + E S E V 1 F + I ξ ξη F ξ + F ξη ξ I + ξη
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1111 Thus, by orollary 36, M # M + E S E V 2 F ξ M + 2 F ξη ξ I + ξη orollary 312 Let M Proof Let ξ E B F B 1 Then 0 B Then M # exists iff E B F B is invertible Put ξ E B F B 1, then M # 0 I F ξ + ξb ξbb + I F ξ + E B ξ 1 E B F ξ 1 F, ξb + +, B + Bξ B + Thus, and ξe B F ξ, ξe B 0, F ξb 0, Bξ BB + F ξ I + F ξ II F F ξ I F ξf F F ξ I ξe B F F F ξ I Associate with Theorem 311, we get the results I I orollary 313 [9] Let M and # exists Then Now, we consider the group inverse of M I 2 # I I + E S E V Here, P AF, W E P BB + Since B A 0 # B A by orollary 26, we have exists iff 0 is invertible B A + I 0 2 π π + # # # B A From the proof of Theorem 311, we have 0 EW E P E W E P A + 0 E I 0, + B + EW E P A E W E P A + 0 + E
FDu, Y Xue / Filomat 28:6 2014, 1103 1112 1112 B + Assume that B #, # exist Then P A π, W E P BB # EW E and P A E W E P A + is invertible iff 0 + E B + E W E P is invertible Noting that E W E P B π E W E P and B # + B π is invertible, we have B + E W E P is invertible iff B + E W E P B π B # + B π BB # + E W E P B π I I E W E P B π is invertible iff I B π I E W E P is invertible by Lemma 21 So, by Lemma 22, we have I B π I E W E P is invertible iff B π I E W E P 0 iff B π P 0 Thus, we get the following theorem by orollary 26: B A Theorem 314 [4, 8, 22] Let M with B 0 #, # exist Then M # exists iff B π A π 0 In this case, B M # # B # 2 A π + B π A # 2 B # A # 0 # 0 1 0 1 0 B Example 315 Let M Put B, D Then M Taking B 1 0 1 0 I D + 1 0 0 1 1 0 1 0 1 1 Then R DF B, W E 1 0 R and DF B B + E W E R F B + DB + B is invertible Thus, by 1 0 Theorem 310, we have M # exists and M # M 2 References [1] Bu, K Zhang, J Zhao, Some results on the group inverse of the block matrix with a sub-block of linear combination or product combination of matrices over skew fields Linear and Multilinear Algebra, 58 8 2010 957 966 [2] Bu, M Li, K Zhang, L Zheng, Group inverse for the block matrices with an invertible subblock Appl Math omput, 215 2009 132 139 [3] J Benítez, XJ Liu, and TP Zhu Additive results for the group inverse in an algebra with applications to block operators Linear Multilinear Algebra, 59 2011 279 289 [4] S L ampbell, D Meyer, Jr, Generalized Inverses of Linear Transformations, Dover, 1991 [5] SL ampbell, Singular Systems of Differential Equations I, Pitman, San Francisco, A, 1980 [6] SL ampbell, Singular Systems of Differential Equations II, Pitman, San Francisco, A, 1980 [7] SL ampbell, The Drazin inverse and systems of second order linear differential equations, Linear and Multilinear Algebra, 14 1983 195 198 [8] ao, Some result of group inverse of partitioned matrix on body J Nat Sci Hei-longjiang Univ, 18 32001 5 7 [9] N astro-gonzález, E Dopazo, Representations of the Drazin inverse for a class of block matrices, Linear Algebra Appl, 400 2005 253 269 [10] N astro-gonzález, JJKoliha, New additive results for the g-drazin inverse, Proc Roy Soc Edinburgh, 134A 2004, 1085 1097 [11] N astro-gonzález, E Dopazo, MFMartlez-Serrano, On thedrazin inverse of the sumof two operators and its application to operator matrices,j Math Anal Appl, 350 2009 207 215 [12] N astro-gonzalez, MF Martlnez-Serrano, Drazin inverse of partitioned matrices in terms of Banachiewicz-Schur forms Linear Algebra Appl, 432 2010 1691 1702 [13] M atral, DD Olesky, P Van Den Driessche, Block representations of the Drazin inverse of a bipartite matrix, Electron J Linear Algebra, 18 2009 98 107 [14] X hen, RE Hartwig, The group inverse of a triangular matrix Linear Algebra Appl, 238 1996 97 108 [15] DS vetković-ilić, A note on the representation for the Drazin inverse of 2 2 block matrices, Linear Algebra Appl, 220 2008 242 248 [16] DS vetković-ilić, DS Djordjević, YWei, Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl, 418 2006 53 61 [17] Deng, YWei, A note on the Drazin inverse of an anti-triangular matrix Linear Algebra Appl, 431 2009 1910 1922 [18] Deng, A note on the Drazin inverses with Banachiewicz-Schur forms Appl Math omput, 213 2009 230 234 [19] F Du, Y Xue, The perturbation of the group inverse under the stable perturbation in a unital ring, Filomat, 27 1 2013 65 74 [20] DS Djordjević, YWei, Additive results for the generalized Drazin inverse, J Aust Math Soc, 73 2002 115 125 [21] R E Hartwig, Block generalized inverses, Arch Ratio Mech Anal, 61 3 1976 197 251 [22] D Meyer, Jr, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev 17 1975 443 464 [23] P Robert, On the group inverse of a linear transformation, J Math Anal Appl, 22 1968 658 669 [24] Y Xue, Stable Perturbations of Operators and Related Topics, World Scientific, 2012 [25] J Zhao, J Bu, Group inverse for the block matrix with two identical subblocks over skew fields Electron J Linear Algebra, 23 2010 63 75