Explicit representations of the Drazin inverse of ... - University of Nis

Explicit representations of the Drazin inverseof block matrix and modified matrix ⋆Dragana S. Cvetković-Ilić a , Jianbiao Chen b Zhaoliang Xu b ∗ ,a Faculty of Mathematics and Sciences University of Nis, Visegradska 33 18000 NisSerbiab Department of Mathematics, Shanghai Maritime University, Shanghai 200135,P.R. ChinaAbstractA new result on the Drazin inverse of 2 × 2 block matrix M = [ ]A BD C , where A andC are square matrices are presented, extended in the case when D = 0, the wellknown representation for the Drazin inverse of M, given by Hartwig, Meyer andRose in 1977, respectively. Using that new result, an explicit representation for theDrazin inverse of a modified matrix P + RS and its generalized matrix P Q + RSare also presented.Key words: Drazin inverse, Modified matrix, Index, Block matrixAMS classification: 15A091 IntroductionLet A ∈ C n×n . The smallest nonnegative integer k such that rank(A k+1 ) =rank(A k ), denoted by Ind(A), is called the index of A. If Ind(A) = k, thereexists a unique matrix A D ∈ C n×n satisfying the following equations [2]A k+1 A D = A k , A D AA D = A D , AA D = A D A. (1.1)⋆ First author is supported by Grant No. 144003 of the Ministry of Science, Technologyand Development, Republic of Serbia. Second and third authors are supportedby the Foundation of Shanghai Education Committee Project (No. 06FZ024 )∗ Corresponding author: zlxu@shmtu.edu.cn(Z. Xu).E-mail address: dragana@pmf.ni.ac.yu(D. Cvetković-Ilić), chenjianbiao@gmail.com(J. Chen).Preprint submitted to Linear and Multilinear Algebra 2 November 2007

A D is called the Drazin inverse of A. In particular, when Ind(A) ≤ 1, the matrixA D is called the group inverse of A and denoted by A # . Clearly, Ind(A)=0if and only if A is nonsingular, and in this case A D = A −1 .The theory of Drazin inverses has seen a substantial growth over the past fewdecades. It is an area which is of great theoretical interest and finds applicationsin many diverse areas, including Statistics, Numerical analysis, Differentialequations, Markov chains, Population models, Cryptography, and Controltheory(cf. [1,2,8,10,13,16,17,19]). One topic of Drazin inverse of considerableinterest concerns the explicit representations for the Drazin inverse of a 2 × 2block matrix (cf. [3,5,11,15,18,22]) and explicit representations for the Drazininverse of the sum of two matrices (cf. [4,12,14,20,21,23]).A sizeable literature devoted to these topics has sprouted. The representationof the Drazin inverse of M = [ ]A B0 C (abbreviated hereafter as Hartwig-Meyer-Rose formula, cf. [2, Theorem 7.7.1], [9], or [18]) is employed frequentlyand plays an essential role, where A and C are square matrices. Therefore,it would be desirable to extend the Hartwig-Meyer-Rose formula to the caseM = [ ]A BD C , when D ≠ 0. Many papers have considered this open problemand each of them offered a formula for the Drazin inverse and specific conditionsfor the 2 × 2 block matrix M = [ ]A BD C to satisfy so that the formula isvalid. Some of those conditions are listed below:(i) rank(M) = rank(A) and Ind(A) = 0, (see [2]);(ii) B = 0 (or D = 0), (see [9],[18]);(iii) BD = 0, CD = 0 and BC = 0 (see [7]);(iv) BD = 0, CD = 0 (or BC = 0) and C is nilpotent, (see [11]);(v) A π B = 0, DA π = 0 and C − DA D B is either nonsingular or zero, (see[22]);(vi) AA π B = 0, DA π B = 0 and C − DA D B is either nonsingular or zero,(see [11]);(vii) AA π B = 0, DA π B = 0, BDAA D = 0 and CDAA D = 0 (see [15]);(viii) A = I and CD = 0 (or BC = 0), (see [5]).In this paper we will give some new conditions for a 2 × 2 matrix M underwhich we obtain an explicit representation of the Drazin inverse of M. Based onthese results, in Section 3 we present some interesting explicit representationsof the Drazin inverse of a modified matrix. Finally, in Section 4 we presentseveral numerical examples showing that there exist matrices for which none2

which implies that⎡⎢MX = XM = ⎣ AD A + X 1 DC D D⎤A D B + X 1 C⎥⎦ . (2.3)C D CNoting thatfrom (2.3) it follows thatDX 1 = 0, DX 2 = 0A D AX 2 D + A D B(C D ) 2 D + X 1 C D D = X 2 DAA D X 1 + A D BC D + X 1 CC D = X 1 ,⎡⎤ ⎡⎤⎢XMX = ⎣ AD A + X 1 D A D B + X 1 C⎥⎢⎦ . ⎣ AD + X 2 D X 1 ⎥⎦C D DC D C (C D ) 2 D C D⎡⎤⎢= ⎣ AD + X 2 D X 1 ⎥⎦ ,(C D ) 2 D C Di.e., XMX = X.Finally, we assert that M k+1 X = M k , for every integer k ≥ Ind(A) +Ind(C) + 2. In fact, by induction on integer k ≥ 2, we have⎡M k ⎢= ⎣ Ak + ∑ ⎤k−2i=0 A i BC k−i−2 ∑D k−1i=0 A i BC k−i−1 ⎥⎦ , (2.4)C k−1 D C knoting that DA = DB = 0. Combining (2.3) and (2.4), we obtainM k+1 X = M k MX⎡⎢= ⎣ Ak + ∑ k−2i=0 A i BC k−i−2 D ∑ ⎤ ⎡⎤k−1i=0 A i BC k−i−1 ⎥ ⎢⎦ ⎣ AD A + X 1 D A D B + X 1 C⎥⎦C k−1 D C k C D D C D C⎡A k A D A + A k X 1 D + A k A D B + A k X 1 C +∑ k−1i=0 A=i BC k−i−1 C D ∑Dk−1i=0 A i BC k−i−1 C D C⎢⎣C k C D DC k C D C⎡⎢= ⎣ Ak + ∑ ⎤k−2i=0 A i BC k−i−2 ∑D k−1i=0 A i BC k−i−1 ⎥⎦ ,C k−1 D C k⎤⎥⎦4

that is to say, M k+1 X = M k , for every integer k ≥ Ind(A)+Ind(C)+2. HenceX = M D and Ind(M) ≤ Ind(A) + Ind(C) + 2. This completes the proof. ✷Theorem 2.2. Let M = [ ]A BD C ∈ C n×n , where A and C are square matrices.If BD = 0 and CD = 0 then⎡⎤M D ⎢= ⎣ AD X 1 ⎥⎦ , (2.5)D(A D ) 2 C D + DX 2where X 1 and X 2 are defined in (2.2). Furthermore, Ind(M) ≤ Ind(A) +Ind(C) + 2.Proof. The proof is similar to the proof of Theorem 2.1.✷3 Explicit representations of the Drazin inverse of a modified matrixand its generalized matrixWe will give, under some conditions several explicit representations of theDrazin inverse of a modified matrix P + RS and its generalized matrix P Q +RS. First, we present two lemmas and then we derive several new resultsconcerning the modified matrix and its generalized matrix.Lemma 3.1. ([6]) Let A ∈ C m×n and B ∈ C n×m . Then (AB) D = A[(BA) D ] 2 B.Lemma 3.2. Let M = [ ]A BD C ∈ C n×n , where A and C are square matrices.Then the following assertions hold⎡⎤If DA = 0 and DB = 0, then (M D ) 2 ⎢= ⎣ (AD ) 2 + X 2 D X 1 ⎥⎦ ; (3.1)(C D ) 3 D (C D ) 2⎡⎤If BD = 0 and CD = 0, then (M D ) 2 ⎢= ⎣ (AD ) 2 X 1 ⎥⎦ . (3.2)D(A D ) 3 (C D ) 2 + DX 2where( lC∑−1X i =X i (A, B, C) = (A D ) i+j+2 BC)(I j − CC D ) (3.3)j=0( lA+ (I − AA D ∑−1) A j B(C D ) i+j+2) i∑− (A D ) j+1 B(C D ) i−j+1 ,j=0j=0for i = 1, 2.5

Proof. After straightforward computation, the above assertions (3.1) and(3.2) follow readily from Theorem 2.1 and Theorem 2.2, respectively. ✷Theorem 3.1. Let P ∈ C m×n , Q ∈ C n×m , R ∈ C m×l and S ∈ C l×m . IfSP QP = 0 and SP QR = 0, then(P Q + RS) D=P ( (QP ) D) 2Q + P X2 SP Q + P X 1 S + R ( (SR) D) 3 ( )SP Q + R (SR)D2S[ ] ⎡ ( ) ⎤2 ⎡ ⎤⎢ (QP )D + X2 SP X 1= P, R ⎣ ( ) 3 ( ) ⎥2⎦⎣ Q ⎦(SR)D SP (SR)D Swhere X i = X i (QP, QR, SR), i = 1, 2 are defined as in (3.3).Proof. By Lemma 3.1, we have( [ ] ⎡ ⎤(P Q + RS) D = P, R ⎣ Q ) D [ ] ( ⎡ ⎤ D ⎡ ⎤)⎦ = P, R ⎣QP QR 2⎦ ⎣ Q ⎦ (3.4)SSP SR SUsing (3.4) and (3.1), we obtain(P Q + RS) D=P ( (QP ) D) 2Q + P X2 SP Q + P X 1 S + R ( (SR) D) 3SP Q + R((SR)D ) 2S,where X i = X i (QP, QR, SR), i = 1, 2 are defined as in (3.3). ✷Theorem 3.2. Let P ∈ C m×n , Q ∈ C n×m , R ∈ C m×l and S ∈ C l×m . IfQRSP = 0 and SRSP = 0, then(P Q + RS) D=P ( (QP ) D) 2Q + RSP X2 S + P X 1 S + RSP ( (QP ) D) 3 ( )Q + R (SR)D2S[ ] ⎡ ( ) ⎤2⎡ ⎤⎢ (QP )DX 1= P, R ⎣SP ( (QP ) D) 3SP X 2 + ( ⎥(SR) D) 2⎦⎣ Q ⎦Swhere X i = X i (QP, QR, SR), i = 1, 2 are defined as in (3.3).Proof. The proof is equivalent to that of Theorem 3.1.✷Applying Theorem 3.1 to the case where Q = I and R = I, we obtain thefollowing explicit representation of the Drazin inverse of matrix S + P , whichis the main result of [12].6

Corollary 3.1.([12]) Let S, P ∈ C m×m . If SP = 0, thenm−1(S + P ) D ∑= (P D ) j+1 S j (I − SS D ) +j=0l∑S −1=j=0(P D ) j+1 S j (I − SS D ) +m−1 ∑j=0l P ∑−1j=0(I − P P D )P j (S D ) i+j(I − P P D )P j (S D ) j+1If we consider Theorem 3.1 in the case where R = P and Q = P k−1 (k ≥ 1)and in the case where Q = I and Theorem 3.2 in the case where R = Q k−1and S = Q and in the case where R = I, we deduce the following:Corollary 3.2. (1) Let P, Q, and S ∈ C m×m . For any integer k ≥ 1, thefollowing assertions hold:(i) If SP k+1 = 0, then(P k + P S) D = (P D ) k + (P S) D + P ( (SP ) D) 2P k−1 + P X 1 S + P X 2 SP k ,(ii) If Q k+1 P = 0, then(P Q + Q k ) D = (Q D ) k + (P Q) D + Q k−1( (QP ) D) 2Q + P Y1 Q + Q k P Y 2 Q,where X i = X i (P k , P k , SP ) and Y i = X i (QP, Q k , Q k ), for i = 1, 2 are definedas in (3.3);(2) Let P ∈ C m×n , R ∈ C m×l , and S ∈ C l×m . Then the following assertionshold:(i) If SP 2 = 0 and SP R = 0, then(P + RS) D = P D + R ( (SR) D) 2S + R((SR)D ) 3SP + P X1 S + P X 2 SP.(ii) If SP R = 0, and P 2 R = 0, then(P + RS) D = P D + R ( (SR) D) 2S + P R((SR)D ) 3S + RY1 P + P RY 2 Pwhere X i = X i (P, R, SR) and Y i = X i (SR, S, P ), i = 1, 2 are defined as in(3.3).4 ExamplesThe following example describes a 2×2 matrix M which does not satisfy anyof the conditions (i)-(viii) that can be found in the papers [2,5,7,9,11,15,18,22],whereas the conditions of Theorem 2.1 are met, which allows us to computeM D .7

⎡ ⎤⎢Example 1. Let M =A B ⎥⎣ ⎦, where A =D C⎡ ⎤⎡ ⎤⎢C =1 1 ⎥⎢⎣ ⎦ and D =1 0 −1 0 ⎥⎣ ⎦.0 01 0 −1 0⎡ ⎤0 1 −1 10 1 −1 1, B =⎢0 1 −1 1⎥⎣ ⎦0 0 0 1⎡ ⎤1 −11 0,⎢1 −1⎥⎣ ⎦0 0It is calculated that⎡ ⎤0 0 0 1A D 0 0 0 1=⎢0 0 0 1⎥⎣ ⎦0 0 0 1⎡ ⎤1 0 0 −1and A π 0 1 0 −1=.⎢0 0 1 −1⎥⎣ ⎦0 0 0 0Now, it is evident that:(i) rank(A) = 2 ≠ rank(M),(ii) B ≠ 0 and D ≠ 0,(iii) CD ≠ 0,(iv) BD ≠ 0,(v) DA π ≠ 0,(vi) C − DA D B = C is neither nonsingular nor zero,(vii) AA π B ≠ 0 ,(viii) CD ≠ 0, BC ≠ 0,so the representations of M D in the literature ([2,7,9,11,18,15,22]) fail toapply, while Theorem 2.1 on the other hand, gives the exact value of M D .Since, C is idempotent, it follows that C D = C and by A D B = 0, we get,8

⎡ ⎤1 11 1X 1 = X 2 = BC =. Using a representation (2.1) in Theorem 2.1,⎢1 1⎥⎣ ⎦0 0⎡⎤2 0 −2 1 1 12 0 −2 1 1 1M D 2 0 −2 1 1 1=.0 0 0 1 0 0⎢2 0 −2 0 1 1⎥⎣⎦0 0 0 0 0 0The following example shows that in the case when none of the conditions(i)-(viii) are satisfied, Theorem 2.2 can also come in handy:Example 2. Let M =⎡ ⎤⎢D =0 0 ⎥⎣ ⎦.−2 1⎡ ⎤⎢ A B ⎥⎣ ⎦, where A =D C⎡ ⎤⎢ 1 1 ⎥⎣ ⎦, B = C =0 0⎡ ⎤⎢ 1 0 ⎥⎣ ⎦ and1 0It can be checked that none of the conditions (i)-(viii) are satisfied and theconditions from Theorem 2.2 are. So, using the representation for M D givenin Theorem 2.2, we get that⎡⎤1 1 −3 0M D 0 0 1 0=.⎢ 0 0 1 0⎥⎣⎦−2 −2 12 0The next example illustrates how useful Theorem 3.1 can be for computingthe Drazin inverse of generalized modified matrices.9

Example 3. Let⎡⎤26 45 −28 15 30−7 −13 11 −23 6M = P Q + RS =−43 −14 29 −34 −72,⎢⎣ −7 −13 11 −23 6 ⎥⎦17 −1 −1 −11 42where⎡ ⎤⎡ ⎤3 10 −6⎡⎤⎡⎤1 2−3 3P =2 3, Q = ⎣−5 5 −5 −5 0⎦ , R =6 6, S = ⎣−2 1 1 1 −7⎦ .5 0 5 0 0−6 −5 3 −5 −5⎢⎣1 2⎥⎢⎦⎣−3 3 ⎥⎦1 2−6 0Then it can be verified that SP QP = 0 and SP QR = 0. Hence, by Theorem3.1 one can find the Drazin inverse of the 5 × 5 matrix M by computing theDrazin inverse of 2 × 2 matrices QP and SR and performing a few otherstraightforward computations. Thus:⎡ ⎤(QP ) D = ⎣−25 −20⎦25 20D⎡ ⎤⎡ ⎤= 1 ⎣−5 −4⎦ , (SR) D = ⎣42 24⎦5 5 478 24D⎡ ⎤= 1 ⎣ −4 4 ⎦144 13 −7which impliesX 1 = ( I − QP (QP ) D) QR ( (SR) D) 3− (QP ) D QR ( (SR) D) 2− ( (QP ) D) 2QR(SR)D⎡⎤= 1 ⎣−229 199 ⎦ ,4320 229 −199X 2 = ( I − QP (QP ) D) QR ( (SR) D) 4− (QP ) D QR ( (SR) D) 3− ( (QP ) D) 2 ( )QR (SR)D2 ( )− (QP )D3QR(SR)D⎡⎤1= ⎣ 38251 −32281⎦ ,3110400 −38251 3228110

noting that (I − QP (QP ) D )QP = 0 and I − SR(SR) D = 0. Now we haveM D =(P Q + RS) D[ ] ⎡ ( ) ⎤2 ⎡ ⎤⎢ (QP )D + X2 SP X 1= P, R ⎣ ( ) 3 ( ) ⎥2⎦⎣ Q ⎦(SR)D SP (SR)D S⎡⎡⎤3 1 0 −61 2 −3 3=⎢2 3 6 6⎥⎣1 2 −3 3 ⎦1 2 −6 0 ⎢⎣⎡− 3554=⎢⎣11− 904936 5760399110800− 3991 − 355910800 10800− 72592− 72592611036827127− 149 − 336798640 540 864017− 1359154 864011− 904936 57601− 473 305336077540533602153559− 22910800 432061− 14310368 207361491080107535760− 2372016687− 778640 1080107535760− 2372059− 130 151994320229− 1994320 432017− 115184 5184⎤.⎥⎦10120736⎤⎡⎤−5 5 −5 −5 0⎢ 5 0 5 0 0⎥⎣−2 1 1 1 −7⎦−6 −5 3 −5 −5⎥⎦Acknowledgments The authors are most grateful to Professor S. Kirklandfor his useful suggestions and to the referee for constructive comments towardsimprovement of the original version of this paper.References[1] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory andApplications, 2nd Edition, Springer Verlag, New York, 2003.[2] S. L. Campbell and C. D. Meyer, Generalized Inverse of LinearTransformations, Pitman, London, 1979, Dover, New York, 1991.[3] N. Castro-González and E. Dopazo, Representations of the Drazin inverse fora class of block matrices, Linear Algebra Appl., 397 (2005) 279-297.[4] N. Castro González, Additive perturbation results for the Drazin inverse, LinearAlgebra Appl., 397 (2005) 279-297.[5] N. Castro-González, E. Dopazo and J. Robles, Formulas for the Drazin inverseof special block matrices, Appl. Math. Comput., 174 (2006) 252-270.11