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A note on the representation for the Drazin inverse of 2×2 block matrices
Linear Algebra and its Applications 429 (2008) 242–248 www.elsevier.com/locate/laa
A note on the representation for the Drazin inverse of 2 × 2 block matrices聻 Dragana S. Cvetkovi´c-Ili´c Department of Mathematics, Faculty of Sciences, University of Niš, P.O. Box 224, Višegradska 33, 18000 Niš, Serbia Received 9 November 2007; accepted 18 February 2008 Available online 1 April 2008 Submitted by R.A. Brualdi
Abstract In this note we consider representations of the Drazin inverse of 2 × 2 block matrices under conditions weaker than those used in recent papers on the subject, in particular in [D.S. Djordjevi´c, P.S. Stanimirovi´c, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (126) (2001) 617–634; R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2 × 2 block matrix, SIAM J. Matrix Anal. Appl. 27 (2006) 757–771]. © 2008 Elsevier Inc. All rights reserved. AMS classification: 15A09 Keywords: Block matrix; Drazin inverse; Additive result
1. Introduction Let A ∈ Cn×n . By R(A), N(A) and rank(A) we denote the range, the null space and the rank of matrix A, respectively. The smallest nonnegative integer k such that rank(Ak+1 ) = rank(Ak ), denoted by ind(A) or iA , is called the index of A. If ind(A) = k, there exists a unique matrix Ad ∈ Cn×n satisfying the following equations: Ak+1 Ad = Ak ,
Ad AAd = Ad ,
AAd = Ad A
and Ad is called the Drazin inverse of A (see [1,4,11,16]). 聻
Supported by Grant No. 144003 of the Ministry of Science, Technology and Development, Republic of Serbia. E-mail addresses: [email protected], [email protected]
0024-3795/$ - see front matter ( 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2008.02.019
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
243
In particular, when ind(A) 1, the matrix Ad is called the group inverse of A and denoted by Clearly, ind(A) = 0 if and only if A is nonsingular, and in this case Ad = A−1 . We denote by Aπ = I − AAd , the projection on N(Ak ) along R(Ak ), where k = ind(A). The Drazin inverse of a square matrix has various applications in singular differential equations and singular difference equations, Markov chains and iterative methods (see [2,4,8,10,14,19]). Campbell and Meyer [4] posed an open to find an explicit representation for the Drazin problem B inverse of a 2 × 2 block matrix M = CA D in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices but their sizes need not be the same. Such representations will, in particular, be very useful to find a general expression for the solutions of the second-order system of the differential equations (see [2,3,13]). Until now, there has been no explicit formula for the Drazin inverse of M in terms of Ad and d D with arbitrary A, B, C and D. However, a general expression for the Drazin inverse of a 2 × 2 block triangular matrix (either B = 0 or C = 0) is presented in the papers of Hartwig et al. [12] and of Meyer and Rose [15]: A# .
Theorem 1.1. If M = A0 d X A Md = , 0 Dd
B D
, where A ∈ Cn×n and D ∈ Cm×m , then (1)
where X = X(A, B, D) =
i D −1
(Ad )n+2 BD n D π +
n=0
i A −1
Aπ An B(D d )n+2 − Ad BD d .
(2)
n=0
Many other papers have considered this open problem and each of them offered a formula for A B the Drazin inverse and specific conditions for the 2 × 2 block matrix M = C D to satisfy so that the formula is valid (see [4,5,10,13,17,18]). In the paper of Djordjevi´c and Stanimirovi´c [10] the representation of M d is given under the conditions: BC = 0,
DC = 0
and
BD = 0,
while in the paper of Hartwig et al. [13] the representation of M d is given under the conditions: BC = 0,
DC = 0
and
D is nilpotent.
In this paper, using an additive result for the Drazininverese proved in [7], we derive formulae A B for the Drazin inverse of a 2 × 2 block matrix M = C D , under the conditions weaker than those in the papers [10,12,13,15]. 2. Results First, we present an additive result for the Drazin inverse proved in [6], which we will be useful in proving our main result. Remark that an additive results for the Drazin inverse of the elements of Banach algebra we can also find in the papers [7,9]. Theorem 2.1. Let P , Q ∈ Cn×n be such that Q2 = 0 and the following conditions are satisfied:
244
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
P π Q = Q,
P QP π = 0.
(3)
Then iP
(P + Q)d = P d +
P i Q(P d )i+2 +
i P −1
i=0
QP i Q(P d )i+3 .
i=0
In the following theorem we derive a formula for the Drazin inverse of block matrix M under some rather cumbersome and complicated conditions but the theorem itself will have a number of useful consequences which will include much simpler conditions. Theorem 2.2. Let M = D π C = C,
A C
B D
, where A ∈ Cn×n and D ∈ Cm×m . If
BCAπ = 0,
DCAπ = 0
(4)
and i D −1
(Ad )n+1 BD n C = 0,
n=0 i D −1
DC(Ad )n+1 BD n D π = 0,
(5)
n=0 i D −1
BC(Ad )n+1 BD n D π = 0,
n=0
then Md =
Ad 0
+
iP X A + d 0 D i=0
i P −1 i=0
0 C
0 0
A 0
B D
B D
i
0 C
i
0 C
0 0 0 0
d A 0
d A 0
X Dd
where X = X(A, B, D) is defined by (2) and iP = ind Proof. We rewrite M = P + Q, where P = d X A d P = , 0 Dd
A 0
B D
X Dd
A 0
i+2
i+3 (6)
, B D
and Q =
.
0 C
0 0
. By Theorem 1.1
where X = X(A, B, D) is defined by (2). Now, we have that the condition P π Q = Q is equivalent to − (AX + BD d )C = 0, D π C = C,
(7)
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
245
while the condition P QP π = 0 is equivalent to BCAπ = 0, DCAπ = 0, −BC(AX + BD d ) = 0,
(8)
d
−DC(AX + BD ) = 0.
(9)
Since d
AX + BD =
i D −1
d n+1
(A )
BD D + n
π
n=0
i A −1
Aπ An B(D d )n+1 ,
n=0
under condition (4), we get that conditions (7)–(9) are equivalent to (5). Hence, if (4) and (5) hold, then the conditions from Theorem 2.1 are satisfied. Now, by Theorem 2.1, we get that M d is represented by (6). Remark that the conditions from (4) are equivalent with the following geometrical conditions: R(C) ⊆ R(D π ) = N(D iD )
and
R(Aπ ) = N(AiA ) ⊆ N(BC) ∩ N(DC).
We have that if BC = 0 and DC = 0 then the conditions of Theorem 2.2 are satisfied and we get a representation of M d . So we can see that the condition of Lemma 2.2 of [13] that D is nilpotent is actually superfluous, as well as the condition BD = 0 from Theorem 5.3 of [10]. B Corollary 2.1. Let M = CA D , where A ∈ Cn×n and D ∈ Cm×m . If BC = 0 and DC = 0, then X Ad , Md = C(Ad )2 Y + D d where X = X(A, B, D) is defined by (2) and Y = CXD d + CAd X. Proof. If BC = 0 and DC = 0, it is evident that i 0 0 0 0 A B = , i 1, C 0 0 0 0 D so d 0 0 X A Md = , + C(Ad )2 CXD d + CAd X 0 Dd where X = X(A, B, D) is defined by (2). Note that Y = CXD d + CAd X =
i D −1
C(Ad )j +3 BD j D π
j =0
−
1 j =0
C(Ad )j +1 B(D d )2−j +
i A −1
CAπ Aj B(D d )j +3 .
j =0
The following theorem presents conditions weaker than those given in Theorem 2.2 under which the representation of M d given by (6) is also valid.
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D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
Theorem 2.3. Let M = ditions is satisfied: (1) (2) (3) (4)
A C
B D
, where A ∈ Cn×n and D ∈ Cm×m . If one of the following con-
D π C = C, BCAπ = 0, DCAπ = 0, Aπ B = B, CAπ = 0, D π C = C, Aπ B = B, D π C = C, BCAπ = 0, DCAπ = 0, Ad BD = 0 = 0, Ad BC = 0, CAd B = 0,
then M d has a representation of the form (6).
Proof. The proof follows directly from Theorem 2.2.
Using that M = P + Q, where P = CA D0 and Q = 00 B0 , we get a similar result as in Theorem 2.2: B Theorem 2.4. Let M = CA D , where A ∈ Cn×n and D ∈ Cm×m . If Aπ B = B,
ABD π = 0,
CBD π = 0
(10)
and i A −1
AB(D d )n+1 CAn Aπ = 0,
n=0 i A −1
CB(D d )n+1 CAn Aπ = 0,
(11)
n=0 i A −1
(D d )n+1 CAn B = 0,
n=0
then Md =
Ad X
+
iP 0 A + C Dd i=0
i P −1 i=0
0 D
0 0
B 0
0 D
A C
i
i
0 0
0 0
B 0 B 0
Ad X
Ad X
where X = X(D, C, A) is defined by (2) and iP = ind
0 Dd 0 Dd
A C
i+2
i+3 (12)
, 0 D
.
Remark that the conditions from (10) are equivalent with the following geometrical conditions: R(B) ⊆ N(AiA ) = R(Aπ ) Corollary 2.2. Let M = then
A C
B D
and
R(D π ) = N(D iD ) ⊆ N(AB) ∩ N(CB).
, where A ∈ Cn×n and D ∈ Cm×m . If AB = 0 and CB = 0,
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
Md =
Ad + Y X
247
B(D d )2 , Dd
where X = X(D, C, A) is defined by (2) and Y = BXAd + BD d X. Theorem 2.5. Let M = ditions is satisfied: (1) (2) (3) (4)
A C
B D
, where A ∈ Cn×n and D ∈ Cm×m . If one of the following con-
Aπ B = B, ABD π = 0, CBD π = 0, D π C = C, BD π = 0, Aπ B = B, D π C = C, Aπ 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. Example The following example describes a 2 × 2 matrix M which does not satisfy the conditions of [10, Theorem 5.3] nor the conditions of [12, Lemma 2.2], whereas the conditions of Theorem 2.2 are met, which allows us to compute M d . B Example. Consider a 2 × 2 block matrix M = CA D , where A = 01 00 , B = D = 01 01 and C = 11 00 . Since BD = / 0 and D is not nilpotent, the mentioned results from [10,12] fail to apply. It is evident that BC = 0 and DC = 0, so we can apply Corollary 2.1, thus obtaining ⎡ ⎤ 1 0 −1 −1 ⎢0 0 0 0⎥ ⎥. Md = ⎢ ⎣1 0 −1 −1⎦ −1 0 2 2
Acknowledgment The author is grateful to the referee for constructive comments towards improvement of the original version of this paper. References [1] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, second ed., Springer-Verlag, New York, 2003. [2] S.L. Campbell, Singular Systems of Differential Equations, Pitman, London, 1980. [3] S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear and Multilinear Algebra 14 (1983) 195–198. [4] S.L. Campbell, C.D. Meyer, Generalized Inverse of Linear Transformations, Pitman, London, 1979, Dover, New York, 1991. [5] N. Castro-González, E. Dopazo, J. Robles, Formulas for the Drazin inverse of special block matrices, Appl. Math. Comput. 174 (2006) 252–270.
248
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
[6] N. Castro-González, Additive perturbation results for the Drazin inverse, Linear Algebra Appl. 397 (2005) 279–297. [7] N. Castro González, J.J. Koliha, New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 1085–1097. [8] X. Chen, R.E. Hartwig, The group inverse of a triangular matrix, Linear Algebra Appl. 237–238 (1996) 97–108. [9] D.S. Cvetkovi´c-Ili´c, D.S. Djordjevi´c, Y. Wei, Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl. 418 (2006) 53–61. [10] D.S. Djordjevi´c, P.S. Stanimirovi´c, On the generalized Drazin inverse and generalized resolvent,Czechoslovak Math. J. 51 (126) (2001) 617–634. [11] M.P. Drazin, Pseudoinverse in associative rings and semigroups, Amer. Math. Monthly 65 (1958) 506–514. [12] R.E. Hartwig, J.M. Shoaf, Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices,Austral. J. Math. 24 (A) (1977) 10–34. [13] R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2 × 2 block matrix, SIAM J. Matrix Anal. Appl. 27 (2006) 757–771. [14] R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207–217. [15] C.D. Meyer, N.J. Rose, The index and the Drazin inverse of block triangular matrices, SIAM J. Appl. Math. 33 (1977) 1–7. [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 Drazin inverse 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, Linear and Multilinear Algebra 45 (1998) 131–146. [19] Y. Wei, X. Li, F. Bu, F. Zhang, Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices – application of perturbation theory for simple invariant subspaces, Linear Algebra Appl. 419 (2006) 765–771.
A note on the representation for the Drazin inverse of 2 × 2 block matrices聻 Dragana S. Cvetkovi´c-Ili´c Department of Mathematics, Faculty of Sciences, University of Niš, P.O. Box 224, Višegradska 33, 18000 Niš, Serbia Received 9 November 2007; accepted 18 February 2008 Available online 1 April 2008 Submitted by R.A. Brualdi
Abstract In this note we consider representations of the Drazin inverse of 2 × 2 block matrices under conditions weaker than those used in recent papers on the subject, in particular in [D.S. Djordjevi´c, P.S. Stanimirovi´c, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (126) (2001) 617–634; R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2 × 2 block matrix, SIAM J. Matrix Anal. Appl. 27 (2006) 757–771]. © 2008 Elsevier Inc. All rights reserved. AMS classification: 15A09 Keywords: Block matrix; Drazin inverse; Additive result
1. Introduction Let A ∈ Cn×n . By R(A), N(A) and rank(A) we denote the range, the null space and the rank of matrix A, respectively. The smallest nonnegative integer k such that rank(Ak+1 ) = rank(Ak ), denoted by ind(A) or iA , is called the index of A. If ind(A) = k, there exists a unique matrix Ad ∈ Cn×n satisfying the following equations: Ak+1 Ad = Ak ,
Ad AAd = Ad ,
AAd = Ad A
and Ad is called the Drazin inverse of A (see [1,4,11,16]). 聻
Supported by Grant No. 144003 of the Ministry of Science, Technology and Development, Republic of Serbia. E-mail addresses: [email protected], [email protected]
0024-3795/$ - see front matter ( 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2008.02.019
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
243
In particular, when ind(A) 1, the matrix Ad is called the group inverse of A and denoted by Clearly, ind(A) = 0 if and only if A is nonsingular, and in this case Ad = A−1 . We denote by Aπ = I − AAd , the projection on N(Ak ) along R(Ak ), where k = ind(A). The Drazin inverse of a square matrix has various applications in singular differential equations and singular difference equations, Markov chains and iterative methods (see [2,4,8,10,14,19]). Campbell and Meyer [4] posed an open to find an explicit representation for the Drazin problem B inverse of a 2 × 2 block matrix M = CA D in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices but their sizes need not be the same. Such representations will, in particular, be very useful to find a general expression for the solutions of the second-order system of the differential equations (see [2,3,13]). Until now, there has been no explicit formula for the Drazin inverse of M in terms of Ad and d D with arbitrary A, B, C and D. However, a general expression for the Drazin inverse of a 2 × 2 block triangular matrix (either B = 0 or C = 0) is presented in the papers of Hartwig et al. [12] and of Meyer and Rose [15]: A# .
Theorem 1.1. If M = A0 d X A Md = , 0 Dd
B D
, where A ∈ Cn×n and D ∈ Cm×m , then (1)
where X = X(A, B, D) =
i D −1
(Ad )n+2 BD n D π +
n=0
i A −1
Aπ An B(D d )n+2 − Ad BD d .
(2)
n=0
Many other papers have considered this open problem and each of them offered a formula for A B the Drazin inverse and specific conditions for the 2 × 2 block matrix M = C D to satisfy so that the formula is valid (see [4,5,10,13,17,18]). In the paper of Djordjevi´c and Stanimirovi´c [10] the representation of M d is given under the conditions: BC = 0,
DC = 0
and
BD = 0,
while in the paper of Hartwig et al. [13] the representation of M d is given under the conditions: BC = 0,
DC = 0
and
D is nilpotent.
In this paper, using an additive result for the Drazininverese proved in [7], we derive formulae A B for the Drazin inverse of a 2 × 2 block matrix M = C D , under the conditions weaker than those in the papers [10,12,13,15]. 2. Results First, we present an additive result for the Drazin inverse proved in [6], which we will be useful in proving our main result. Remark that an additive results for the Drazin inverse of the elements of Banach algebra we can also find in the papers [7,9]. Theorem 2.1. Let P , Q ∈ Cn×n be such that Q2 = 0 and the following conditions are satisfied:
244
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
P π Q = Q,
P QP π = 0.
(3)
Then iP
(P + Q)d = P d +
P i Q(P d )i+2 +
i P −1
i=0
QP i Q(P d )i+3 .
i=0
In the following theorem we derive a formula for the Drazin inverse of block matrix M under some rather cumbersome and complicated conditions but the theorem itself will have a number of useful consequences which will include much simpler conditions. Theorem 2.2. Let M = D π C = C,
A C
B D
, where A ∈ Cn×n and D ∈ Cm×m . If
BCAπ = 0,
DCAπ = 0
(4)
and i D −1
(Ad )n+1 BD n C = 0,
n=0 i D −1
DC(Ad )n+1 BD n D π = 0,
(5)
n=0 i D −1
BC(Ad )n+1 BD n D π = 0,
n=0
then Md =
Ad 0
+
iP X A + d 0 D i=0
i P −1 i=0
0 C
0 0
A 0
B D
B D
i
0 C
i
0 C
0 0 0 0
d A 0
d A 0
X Dd
where X = X(A, B, D) is defined by (2) and iP = ind Proof. We rewrite M = P + Q, where P = d X A d P = , 0 Dd
A 0
B D
X Dd
A 0
i+2
i+3 (6)
, B D
and Q =
.
0 C
0 0
. By Theorem 1.1
where X = X(A, B, D) is defined by (2). Now, we have that the condition P π Q = Q is equivalent to − (AX + BD d )C = 0, D π C = C,
(7)
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
245
while the condition P QP π = 0 is equivalent to BCAπ = 0, DCAπ = 0, −BC(AX + BD d ) = 0,
(8)
d
−DC(AX + BD ) = 0.
(9)
Since d
AX + BD =
i D −1
d n+1
(A )
BD D + n
π
n=0
i A −1
Aπ An B(D d )n+1 ,
n=0
under condition (4), we get that conditions (7)–(9) are equivalent to (5). Hence, if (4) and (5) hold, then the conditions from Theorem 2.1 are satisfied. Now, by Theorem 2.1, we get that M d is represented by (6). Remark that the conditions from (4) are equivalent with the following geometrical conditions: R(C) ⊆ R(D π ) = N(D iD )
and
R(Aπ ) = N(AiA ) ⊆ N(BC) ∩ N(DC).
We have that if BC = 0 and DC = 0 then the conditions of Theorem 2.2 are satisfied and we get a representation of M d . So we can see that the condition of Lemma 2.2 of [13] that D is nilpotent is actually superfluous, as well as the condition BD = 0 from Theorem 5.3 of [10]. B Corollary 2.1. Let M = CA D , where A ∈ Cn×n and D ∈ Cm×m . If BC = 0 and DC = 0, then X Ad , Md = C(Ad )2 Y + D d where X = X(A, B, D) is defined by (2) and Y = CXD d + CAd X. Proof. If BC = 0 and DC = 0, it is evident that i 0 0 0 0 A B = , i 1, C 0 0 0 0 D so d 0 0 X A Md = , + C(Ad )2 CXD d + CAd X 0 Dd where X = X(A, B, D) is defined by (2). Note that Y = CXD d + CAd X =
i D −1
C(Ad )j +3 BD j D π
j =0
−
1 j =0
C(Ad )j +1 B(D d )2−j +
i A −1
CAπ Aj B(D d )j +3 .
j =0
The following theorem presents conditions weaker than those given in Theorem 2.2 under which the representation of M d given by (6) is also valid.
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D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
Theorem 2.3. Let M = ditions is satisfied: (1) (2) (3) (4)
A C
B D
, where A ∈ Cn×n and D ∈ Cm×m . If one of the following con-
D π C = C, BCAπ = 0, DCAπ = 0, Aπ B = B, CAπ = 0, D π C = C, Aπ B = B, D π C = C, BCAπ = 0, DCAπ = 0, Ad BD = 0 = 0, Ad BC = 0, CAd B = 0,
then M d has a representation of the form (6).
Proof. The proof follows directly from Theorem 2.2.
Using that M = P + Q, where P = CA D0 and Q = 00 B0 , we get a similar result as in Theorem 2.2: B Theorem 2.4. Let M = CA D , where A ∈ Cn×n and D ∈ Cm×m . If Aπ B = B,
ABD π = 0,
CBD π = 0
(10)
and i A −1
AB(D d )n+1 CAn Aπ = 0,
n=0 i A −1
CB(D d )n+1 CAn Aπ = 0,
(11)
n=0 i A −1
(D d )n+1 CAn B = 0,
n=0
then Md =
Ad X
+
iP 0 A + C Dd i=0
i P −1 i=0
0 D
0 0
B 0
0 D
A C
i
i
0 0
0 0
B 0 B 0
Ad X
Ad X
where X = X(D, C, A) is defined by (2) and iP = ind
0 Dd 0 Dd
A C
i+2
i+3 (12)
, 0 D
.
Remark that the conditions from (10) are equivalent with the following geometrical conditions: R(B) ⊆ N(AiA ) = R(Aπ ) Corollary 2.2. Let M = then
A C
B D
and
R(D π ) = N(D iD ) ⊆ N(AB) ∩ N(CB).
, where A ∈ Cn×n and D ∈ Cm×m . If AB = 0 and CB = 0,
D.S. Cvetkovi´c-Ili´c / Linear Algebra and its Applications 429 (2008) 242–248
Md =
Ad + Y X
247
B(D d )2 , Dd
where X = X(D, C, A) is defined by (2) and Y = BXAd + BD d X. Theorem 2.5. Let M = ditions is satisfied: (1) (2) (3) (4)
A C
B D
, where A ∈ Cn×n and D ∈ Cm×m . If one of the following con-
Aπ B = B, ABD π = 0, CBD π = 0, D π C = C, BD π = 0, Aπ B = B, D π C = C, Aπ 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. Example The following example describes a 2 × 2 matrix M which does not satisfy the conditions of [10, Theorem 5.3] nor the conditions of [12, Lemma 2.2], whereas the conditions of Theorem 2.2 are met, which allows us to compute M d . B Example. Consider a 2 × 2 block matrix M = CA D , where A = 01 00 , B = D = 01 01 and C = 11 00 . Since BD = / 0 and D is not nilpotent, the mentioned results from [10,12] fail to apply. It is evident that BC = 0 and DC = 0, so we can apply Corollary 2.1, thus obtaining ⎡ ⎤ 1 0 −1 −1 ⎢0 0 0 0⎥ ⎥. Md = ⎢ ⎣1 0 −1 −1⎦ −1 0 2 2
Acknowledgment The author is grateful to the referee for constructive comments towards improvement of the original version of this paper. References [1] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, second ed., Springer-Verlag, New York, 2003. [2] S.L. Campbell, Singular Systems of Differential Equations, Pitman, London, 1980. [3] S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear and Multilinear Algebra 14 (1983) 195–198. [4] S.L. Campbell, C.D. Meyer, Generalized Inverse of Linear Transformations, Pitman, London, 1979, Dover, New York, 1991. [5] N. Castro-González, E. Dopazo, J. Robles, Formulas for the Drazin inverse of special block matrices, Appl. Math. Comput. 174 (2006) 252–270.
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