A note of the representation of the Drazin inverse of 2x2 block matrix

A note on the representation for theDrazin inverse of 2 × 2 block matricesDragana S. Cvetković-IlićAbstractIn this note [ the representations ] of the Drazin inverse of a 2×2 blockA Bmatrix M = , where A and D are square matrices, have beenC Drecently developed under some assumptions. We derive formulae forthe Drazin inverse of a block matrix M under conditions weaker thanthose in the papers [9], [11], [12] and [14].2000 Mathematics Subject Classification: 15A09 Key words: block matrix;Drazin inverse; additive result;1 IntroductionLet A ∈ C n×n . By R(A), N (A) and rank(A) we denote the range, the nullspace and the rank of matrix A, respectively. The smallest nonnegativeinteger k such that rank(A k+1 ) = rank(A k ), denoted by ind(A) or i A , iscalled the index of A. If ind(A) = k, there exists a unique matrix A d ∈ C n×nsatisfying the following equationsA k+1 A d = A k , A d AA d = A d , AA d = A d A,and A d is called the Drazin inverse of A (see [1, 4, 10, 16]).In particular, when ind(A) ≤ 1, the matrix A d is called the group inverseof A and denoted by A # . Clearly, ind(A) = 0 if and only if A is nonsingular,and in this case A d = A −1 . We denote by A π = I − AA d , the projection onN (A k ) along R(A k ), where k = ind(A).Supported by Grant No. 144003 of the Ministry of Science, Technology and Development,Republic of Serbia1

In this paper, using an additive result for the Drazin inverese provedin [6],[we derive]formulae for the Drazin inverse of a 2 × 2 block matrixA BM = , under the conditions weaker than those in the papers [9],C D[11], [12] and [14].2 ResultsFirst, we present an additive result for the Drazin inverse proved in [6],which we will be useful in proving our main result.Theorem 2.1 Let P, Q ∈ C n×n be such that Q dconditions are satisfied= 0 and the followingP π Q = Q, P QP π = 0. (3)Then,∞∑(P + Q) d = P d + (P + Q) n Q(P d ) n+2 .n=0In the following theorem we derive a formula for the Drazin inverse ofblock-matrix M under some rather cumbersome and complicated conditionsbut the theorem itself will have a number of useful consequences which willinclude much simpler conditions.[ A BTheorem 2.2 Let M =C Dand], where A ∈ C n×n and D ∈ C m×m . IfD π C = C, BCA π = 0, DCA π = 0 (4)i D −1∑(A d ) n+1 BD n C = 0,n=0i∑D −1n=0i∑D −1n=0DC(A d ) n+1 BD n D π = 0,BC(A d ) n+1 BD n D π = 0,(5)3

thenM d =[ ] AdX0 D d +∞∑[ A BC Dn=0] n []0 0C(A d ) n+2 ∑ n+2i=1 C(Ad ) i−1 X(D d ) n+2−i , (6)where X = X(A, B, D) is defined by (2).Proof. We rewrite M = P + Q, where P =By Theorem 1.1,P d =[ ] AdX0 D d ,[ A B0 D]and Q =[ 0 0C 0where X = X(A, B, D) is defined by (2). Now, we have that the conditionP π Q = Q is equivalent to−(AX + BD d )C = 0, (7)D π C = Cwhile the condition P QP π = 0 is equivalent toSince,AX + BD d =i D −1BCA π = 0, DCA π = 0,−BC(AX + BD d ) = 0, (8)−DC(AX + BD d ) = 0. (9)∑(A d ) n+1 BD n D π +n=0i∑A −1n=0A π A n B(D d ) n+1 ,under the condition (4), we get that conditions (7), (8) and (9) are equivalentto (5). Hence, if (4) and (5) hold, then the conditions from the Theorem 2.1are satisfied. Now, by Theorem 2.1].M d = P d +=∞∑M n Q(P d ) n+2n=0[ ] AdX0 D d +∞∑[ A BC Dn=0] n []0 0C(A d ) n+2 ∑ n+2i=1 C(Ad ) i−1 X(D d ) n+2−i .□4

Note thatn+2∑S n = C(A d ) i−1 X(D d ) n+2−i =i=1i∑D −1+j=0i∑A −1j=0n+2∑C(A d ) n+j+3 BD j D π −j=0CA π A j B(D d ) n+j+3C(A d ) j B(D d ) n+3−j .We have that if BC = 0 and DC = 0 then the conditions of Theorem2.2 are satisfied and we get a representation of M d . So we can see that thecondition of Lemma 2.2 of [12] that D is nilpotent is actually superfluous,as well as the condition BD = 0 from Theorem 5.3 of [9].[ ] A BCorollary 2.1 Let M = , where A ∈ CC Dn×n and D ∈ C m×m . IfBC = 0 and DC = 0, then[M d A=d]XC(A d ) 2 Y + D d ,where X = X(A, B, D) is defined by (2) and Y = CXD d + CA d X.Proof. If BC = 0 and DC = 0, it is evident that[ ] n []A B 0 0C D C(A d ) n+2 ∑ n+2i=1 C(Ad ) i−1 X(D d ) n+2−i = 0, n ≥ 1,so,M d =[ ] [AdX0 D d +0 0C(A d ) 2 CXD d + CA d Xwhere X = X(A, B, D) is defined by (2). Note thatY = CXD d + CA d X =−1∑j=0i∑D −1j=0C(A d ) j+3 BD j D π],i∑A −1C(A d ) j+1 B(D d ) 2−j + CA π A j B(D d ) j+3 .□The following theorem presents conditions weaker than those given inTheorem 2.2 under which the representation of M d given by (6) is alsovalid.j=05

[ A BTheorem 2.3 Let M =C Done of the following conditions is satisfied:], where A ∈ C n×n and D ∈ C m×m . If(1) D π C = C, BCA π = 0, DCA π = 0, A π B = B,(2) CA π = 0, D π C = C, A π B = B,(3) D π C = C, BCA π = 0, DCA π = 0, A d BD = 0 = 0, A d BC = 0,CA d B = 0,then M d has a representation of the form (6).Proof. The proof follows directly from the Theorem 2.2.[ ] A 0Using that M = P + Q, where P =and Q =C Dget a similar result as in Theorem 2.2:[ A BTheorem 2.4 Let M =C Dand[ 0 B0 0], we], where A ∈ C n×n and D ∈ C m×m . IfA π B = B, ABD π = 0, CBD π = 0 (10)i∑A −1n=0i∑A −1n=0i A −1AB(D d ) n+1 CA n A π = 0,CB(D d ) n+1 CA n A π = 0,∑(D d ) n+1 CA n B = 0,n=0(11)thenM d =[ ] Ad0X D d +∞∑[ A BC Dn=0] n [ ∑ n+1i=0 B(Dd ) i X(A d ) n+1−i B(D d ) n+20 0],(12)where X = X(D, C, A) is defined by (2).6

[ A BCorollary 2.2 Let M =C DAB = 0 and CB = 0, thenM d =], where A ∈ C n×n and D ∈ C m×m . If[ A d + Y B(D d ) 2 ]X D d ,where X = X(D, C, A) is defined by (2) and Y = BXA d + BD d X.[ ] A BTheorem 2.5 Let M = , where A ∈ CC Dn×n and D ∈ C m×m . Ifone of the following conditions is satisfied:(1) A π B = B, ABD π = 0, CBD π = 0, D π C = C,(2) BD π = 0, A π B = B, D π C = C,(3) A π B = B, ABD π = 0, CBD π = 0, D d CA = 0 = 0, D d CB = 0,BD d C = 0,then M d can be represented as in (12).3 ExampleThe following example describes a 2 × 2 matrix M which does not satisfyconditions from Theorems 5.3 and Lemma 2.2, respectively of [9] and [11],whereas the conditions of Theorem 2.2 are met, which allows us to computeM D .[ ] A BExample. Consider a 2 × 2 block matrix M =, whereC D[ ][ ] [ ]1 01 11 0A = , B = D = and C = . Since BD ≠ 0 and0 00 0−1 0D is not nilpotent, the mentioned results from [9] and [11] fail to apply. Itis evident that BC = 0 and DC = 0, so we can apply Corollary 2.1, thusobtaining⎡⎤1 0 −1 −1M d = ⎢ 0 0 0 0⎥⎣ 1 0 −1 −1 ⎦ .−1 0 2 27

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[14] C.D. Meyer and N.J. Rose, The index and the Drazin inverse of blocktriangular matrices, SIAM J. Appl. Math., 33 (1977) 1-7.[15] X. Li and Y. Wei, An expression of the Drazin inverse of a perturbedmatrix, Appl. Math. Comput., 153 (2004) 187-198.[16] G. Wang, Y. Wei, S. Qiao, Generalized inverses: theory and computations,Science Press, 2003.[17] X. Li, Y. Wei, A note on the representations for the Drazininverse of 2 × 2 block matrices, Linear Algebra Appl., (2007),doi:10.1016/j.laa.2007.01.005.[18] Y. Wei, Expression for the Drazin inverse of a 2×2 block matrix, Linearand Multilinear Algebra, 45 (1998) 131-146.[19] Y. Wei, X. Li, F. Bu, F. Zhang, Relative perturbation bounds for theeigenvalues of diagonalizable and singular matrices-application of perturbationtheory for simple invariant subspaces, Linear Algebra Appl.,419 (2006) 765–771.Address:Dragana S. Cvetković-Ilić:Department of Mathematics, Faculty of Sciences, University of Niš, P.O.Box 224, Višegradska 33, 18000 Niš, SerbiaE-mail: dragana@pmf.ni.ac.yu gagamaka@ptt.yu9