Linear Algebra and its Applications

Linear Algebra and its Applications 432 21 1691 172 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Drazin inverse of partitioned matrices in terms of anachiewicz Schur forms N. Castro-González, M.F. Martínez-Serrano Departamento de Matemática Aplicada, Facultad de nformática, Universidad Politécnica de Madrid, 2866 oadilla del Monte, Madrid, Spain A R T C L E N F O A S T R A C T Article history: Received 28 April 29 Accepted 19 November 29 Available online 29 December 29 Submitted by S. Kirkland AMS classification: 15A9 15A3 65F2 Keywords: anachiewicz Schur form Drazin inverse nner inverse Schur generalized complement A Let M C D be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A D. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S D CA D, group invertible, can be expressed in terms of a matrix in the anachiewicz Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rankm ranka D + ranks D, and give conditions under which the group inverse of M exists and a formula for its computation. 29 Elsevier nc. All rights reserved. 1. ntroduction Let C m,n be the set of m n complex matrices. Let us recall that a matrix A C n,m is an inner inverse of a given matrix A C m,n if AA A A. Also, a matrix X C n,m is a generalized reflexive inverse of A inner and outer inverse if AXA A and XAX X. The research is partially supported by Project MTM27-67232, Ministerio de Educación y Ciencia" of Spain. Corresponding author. E-mail addresses: nieves@fi.upm.es N. Castro-González, fmartinez@fi.upm.es M.F. Martínez-Serrano. 24-3795/$ - see front matter 29 Elsevier nc. All rights reserved. doi:1.116/j.laa.29.11.24

1692 N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 Now, let A C m,m. Let us define the index of A, inda, as the smallest non-negative integer r such that ranka r ranka r+1.finda r, the Drazin inverse of A is the unique matrix X C m,m satisfying the relations XAX X, AX XA, A k+1 X A k for all k r. 1.1 The Drazin inverse of A will be specified by A D. f inda, then A is nonsingular and the solution to 1.1isA D A 1.findA 1, then A D is a generalized reflexive inverse of A, and it is called the group inverse of A, denoted by A. We will denote by A π the eigenprojection of A corresponding to the eigenvalue, which is given by A π AA D. For the theory of generalized inverses and its applications, we refer the reader to [2,4]. n the following proposition, we have compiled some basic facts about the Drazin inverse, which will be used throughout the paper. We will denote by RA and NA the range and the null space of A, respectively. Proposition 1.1. Let A, C n,n with inda r and let K A 2 A D. Then a RA D RA r and NA D N A r. b RA π NA r and NA π RA r. c indk 1, K D A D, RK RA D and NK NA D. d f P is nonsingular and PAP 1, then D PA D P 1. e f r >, then there exists a nonsingular matrix P such that we can write A in the core-nilpotent block form A1 A P P 1, A A 1 C k,k nonsingular, k ranka r,a r 2, 1.2 2 and, relative to this form, we have A D A 1 P 1 P 1, A π P P 1. Moreover, if inda 1, then A 2 in 1.2 and so A π A AA π. Throughout this paper we consider M C m+n,m+n partitioned as A M, A C C D m,m,d C n,n. 1.3 Representations for the Drazin inverse for block matrices were given in the literature under certain conditions [3,6,7,1,17]. n recent papers [1,15] necessary and sufficient conditions were derived for a partitioned matrix to have several generalized inverses, including inner, reflexive and Moore Penrose inverse, with anachiewicz Schur form. We recall that, if A denotes a generalized inverse of A, then the generalized Schur complement of A in M is defined as S D CA, and we say that the generalized inverse of M has the anachiewicz Schur form when it is expressible in the form M A + A S CA A S S CA S. The Schur complement is a basic tool in many areas of matrix analysis [18]. n the sequel we consider the Drazin inverse of A and the generalized Schur complement S D CA D. Under the condition that S is nonsingular, the following result is known [16]. Lemma 1.2. Let M be partitioned as in 1.3 and let S be as in 1.4. f S is nonsingular, A π and CA π, then 1.4

N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 1693 M D A D + A D S 1 CA D S 1 CA D A D S 1 S 1. 1.5 The case in which the generalized Schur complement is equal to zero has been studied in the literature. n [4, Lemma 3.3.1 and Theorem 7.7.7], it was proved that if A is nonsingular, then rankm ranka if and only if S. n this situation, the group inverse of M exists if and only if + A 1 CA 1 is nonsingular, in which case M CA 1 WAW 1 A 1, W + A 1 CA 1. 1.6 This representation was very useful for the study of the perturbation of the Drazin inverse in [5]. n [12], under the assumptions that A π, CA π and S, it was shown that the Drazin inverse of M is expressible in the form M D CA D A [ WA D ] 2 A D, W AA D + A D CA D. 1.7 Hartwig et al. in [8, Theorems 3.1 and 4.1] gave expressions for the Drazin inverse of M in the cases when S is nonsingular and S, under conditions CA π and AA π. The formulas showed therein involve a matrix in the form 1.5 for the case when S is nonsingular and a matrix in the form 1.7 for the case when S. The paper is organized as follows. n Section 2, we develop conditions under which the Drazin inverse of a partitioned matrix as in 1.3 having group invertible generalized Schur complement, can be expressed in terms of the anachiewicz-schur form of M and its powers. n Section 3, we deal with partitioned matrix satisfying the rank formula rankm ranka D + ranks D, and we give conditions under which the group inverse of M exists and a formula for its computation. We emphasize that when the rank formula holds, then S will be group invertible. From our results we derive the characterization of the group invertibility of M for the cases in which A is nonsingular, and S is nonsingular. n our development we will use the following lemma. t is well-known that if A A, then A + D A D + D. An extension of this additive result under the one side condition A was givenin[9]. Lemma 1.3. Let A C n,n and let C n,n be nilpotent of index r. i f A, then A + D r 1 i AD i+1 i. ii f A, then A + D r 1 i i A D i+1. n order to establish rank equalities for block matrices, we need some rank formulas [11]. Lemma 1.4. Let A C m,p, C m,k,c C n,p,d C n,k,g C p,k. Then i rankag rankg dimrg N A. ii A rank C rank + rankc + rank[m A p C C]. iii A rank ranka + rank. C D m AA C p A A D CA 2. Drazin inverse of block matrices in terms of anachiewicz Schur forms n this section we address the problem of developing conditions under which the Drazin inverse of a partitioned matrix as in 1.3 can be obtained by a formula which involves the anachiewicz Schur form. First, we will derive a formula under conditions A π and S π C, with A and S being

1694 N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 group invertible. Secondly, we will extend this result under less restrictive conditions. n particular we recover the case AA π, CA π and S π C. We will use the notation of the following lemma. Lemma 2.1. Let M be partitioned as in 1.3 and let S be as in 1.4. Let us introduce the nonsingular matrix A U D S D C + S D CA D, U 1 + A D S D C A D S D. 2.1 C f CA π, then UMU 1 A + A D S π C A π S D C S D CA + A D S π C + S π C A D S + A π S + S D CA D. 2.2 S The following theorem is an extension of [13, Theorem 3.3]. Theorem 2.2. Let M be partitioned as in 1.3 and let S be as in 1.4. f inda 1, inds 1,A π and S π C, then M D A + A S CA S CA A S A S CA π S S CA π A S π. 2.3 Proof. First, we observe that if inds, then S is nonsingular. n this case, the expression 2.3 can be derived from [8, Theorem 3.1]. n the sequel, we assume that inds 1. Under assumptions of this theorem, the expression 2.2 reduces to UMU 1 A A S : M, 2.4 S CA S + S CA S where U and U 1 are defined as in 2.1. y Proposition 1.1 e, we have S S1 Q Q 1, where S 1 is nonsingular. Now write, relative to the above decomposition, Q 1 2 and Q 1 C C 1 C. 2 Then Q 1 M Q A A 1 S 1 S 1 1 C 1A S 1 + S 1 1 C 1A 1 S 1. The Schur complement of A in the 2 2 block submatrix of the matrix on the right hand side is equal to S 1 and, thus, it is nonsingular. Since A π A and AA π we can apply Lemma 1.2 to obtain the Drazin inverse of the 2 2 block submatrix and, consequently, we get A + A 2 1 S 1 M D 1 C 1AA A 2 1 Q S 1 2 1 C1 AA S 1 1 Q 1 A + A 2 S CAA A 2 SS S 2 CAA S. Finally, in view of 2.4 and on account of Proposition 1.1 d, we compute M D U A + A 2 S CAA A 2 SS A S 2 CAA S S C + S CA

N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 1695 A A 2 S CA π A 2 S π S 2 CA π S and M D U 1 + A S C MD U S C which gives us the desired result 2.3. A A A 2 S CA π A 2 S π S 2 CA π S, The following particular case recovers [13, Theorem 3.2]. Corollary 2.3. Let M be partitioned as in 1.3 and let S be as in 1.4. f inda 1, inds 1,A π, CA π, S π C and S π, then indm 1 and M A + A S CA S CA A S S Using the following lemma we can prove the counterpart of Theorem 2.2.. Lemma 2.4. Let M be partitioned as in 1.3 and let S be as in 1.4. Let us introduce the nonsingular matrix S V D CA D + CA D S D, V 1 + S D CA D S D CA D. 2.5 f CA π, then V 1 MV A + S π CA D S D CA π SCA D + CA π A + S π CA D S D + S π S + SCA D S D. Theorem 2.5. Let M be partitioned as in 1.3 and let S be as in 1.4. f inda 1, inds 1,CA π and S π, then M D A π S CA S π CA A π S A + A S CA S CA A S S. 2.6 Now, we state the main result of this section. The significance of the assumptions in the theorem will be shown in the next section. Theorem 2.6. Let M be partitioned as in 1.3 and let S be as in 1.4 with inds 1. f AA π, CA π, S π CA π and W + A D S π CA D is nonsingular, then [ M D A + D S π C A π + W A π S π C A π S π CA S π C S π CA D R + D ] R 2 A D S π R r 1 + i R i+1 CA π A i, 2.7 where r inda and A R D + A D S CA D S CA D A D S W 1 S. 2.8

1696 N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 Proof. With the notation W + A D S π CA D, using CA π and S π CA π, from 2.2 we obtain UMU 1 WA 2 A D A D S AA π S CWA 2 A D S + S CA D + S S CAA π 2.9 + : X + Y + Z. A π S C A π S π C Under the assumptions of this theorem, we have X + YZ. On the other hand, Z 2 and Z 3. 2.1 y Lemma 1.3 ii A π S π C UM D U 1 X + Y D + Z X + Y D 2 + Z 2 X + Y D 3. 2.11 Now, it is clear that YX. Since AA π is nilpotent of index r, Y is also nilpotent of index r. y Lemma 1.3 i, r 1 X + Y D X D i+1 Y i. i Moreover, since YX D, we get r 1 [X + Y D ] k X D i+k Y i k 1. 2.12 i y substituting 2.12 in2.11 weget M D [ U 1 + ZX D + Z 2 X D 2] r 1 X D + X D i Y i U. 2.13 i1 To obtain X D we will apply Corollary 2.3. Since W + A D S π CA D is nonsingular, using Proposition 1.1 c, we can easily prove that WA 2 A D D A D W 1, WA 2 A D π A π and indwa 2 A D 1. We note that the Schur complement of WA 2 A D in X is equal to S and the conditions in Corollary 2.3 hold for this partitioned matrix. Thus, X D A D W 1 + A D W 1 A D S CAA D A D W 1 A D SS S 2 CAA D S. 2.14 With U and U 1 defined as in 2.1 and R defined as in 2.8, using 2.14, 2.9 and 2.1 we compute U 1 X D i U R i A D S CA π A D S π S CA π, i 1, U 1 Y i A π A i U, i 1, U 1 A ZU D S π C A π A D S π CA D S π C S π CA D, U 1 Z 2 A π S π C A π S π CA π U. y substituting the above computations in 2.13 and rearranging terms we obtain the formula 2.7.

N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 1697 We can now state the following corollary, which recovers the result of [8, Theorem 3.1] for the case when S is nonsingular. Corollary 2.7. Let M be partitioned as in 1.3 and let S be as in 1.4 with inds 1. f AA π, CA π and S π C, then [ M D A π ] + R R A D S π r 1 + R i+1 CA π A i, where r inda and R is defined as in 2.8 with W. Under the assumptions of Theorem 2.6 with AA π, S π CA π " replaced by CAA π, A π S π ", and using the Lemma 2.4, the further result may be proved in much the same way as Theorem 2.6. Theorem 2.8. Let M be partitioned as in 1.3 and let S be as in 1.4 with inds 1. f CAA π, CA π, A π S π and W + A D S π CA D is nonsingular, then r 1 M D A S π CA D + i A π R i+1 R i [ S + π CA D S π R CA π + W CA D S π + R 2 S π CA π ] CA D S π CA π, where r inda and W 1 A R D + A D S CA D A D S S CA D S. 2.15 Corollary 2.9. Let M be partitioned as in 1.3 and let S be as in 1.4 with inds 1. f CAA π, CA π and S π, then r 1 M D A S π CA D + i A π ] R i+1 R [ + R CA π, i i where r inda and R is defined as in 2.15 with W. 3. Group invertibility of partitioned matrices under block-rank condition n [14], necessary and sufficient conditions for the existence of an inner inverse A such that rankm ranka + rankd CA were given. When we focus attention on the Drazin inverse, it seems natural to consider the block-rank condition rankm ranka D + ranks D. Here we establish a characterization for further use. Theorem 3.1. Let M be partitioned as in 1.3 and let S be as in 1.4. Then rankm ranka D + ranks D if and only if the following conditions hold A π A S D CA π, A π S π, S π CA π and SS π. 3.1 Proof. Let us introduce the matrices AA F π + S D C C SS π, G A 2 A D S 2 S D.

1698 N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 Then, using Proposition 1.1c rank M rankm + ranka D + ranks D. 3.2 G On the other hand, using that matrix multiplications by nonsingular matrices do not change the rank of the matrix we get A M rank rank CA D CA D SS D C D G A 2 A D SS D CA D SS D SS D CA D S π S 2 S D A A 2 A D CA π S CAA D S 2 S D rank A 2 A D SS D CA π S 2 S D SS D CAA D AA D S D C A π 3.3 S D C AA π + S D C A 2 A D C SS π S 2 S D rank A 2 A D F G rank. G S 2 S D We note that G A D S D GG F G G is an inner inverse of G and GG G A G π S π. Further, A π A S D CA π A π S π S π CA π SS π : N. y Lemma 1.4 ii we obtain F G rank 2rankG + rankn 2rankA D + 2rankS D + rankn. 3.4 G Using 3.2 3.4 we obtain rankm ranka D + ranks D + rankn. Hence rankm ranka D + ranks D if and only if rankn or, equivalently, conditions 3.1 hold. Corollary 3.2. Let M be partitioned as in 1.3 and let S be as in 1.4. f S is nonsingular, then rankm ranka D + ranks A π A S 1 CA π. Corollary 3.3. Let M be partitioned as in 1.3 and let S be as in 1.4. Then rankm ranka D A π,ca π,s and AA π. Remark 3.4. Let r inda. A geometrical reformulation of conditions 3.1 is as follows: N S A S D CN A r RA r, C N A r RS and RS RS 2.

N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 1699 We can now state the main result of this section. Theorem 3.5. Let M be partitioned as in 1.3 and let S be as in 1.4. fca π and rankm ranka D + ranks D, then indm 1 if and only if W + A D S π CA D is nonsingular. n this case the group inverse of M has the form [ M A R + D S π C A π ] + W S π C S π CA D R 2 A D S CA π A D S π S CA π, 3.5 where R is defined as in 2.8. Proof. First, note that under assumption rankm ranka D + ranks D, conditions 3.1 also hold. From AA π A π S D CA π and CA π it follows that AA π and CAA π. Further, using U 1 and V 1 defined as in 2.1 and 2.5, respectively, we obtain rankv 1 MMU 1 AA D A S CA π S π A A π S CAA D A π rank CA π S S π C S rank A 3 A D + AA D S π CAA D S 2 ranka 3 A D + AA D S π CAA D + ranks. Hence, since the rank is invariant under matrix multiplications by nonsingular matrices, rankm 2 ranka D + ranks if and only if ranka 3 A D + AA D S π CAA D ranka D. The latter relation holds if and only if RA 3 A D NAA D + AA D S π CA D 2 {}, by Lemma 1.4 i. Which is equivalent to the fact that A π + AA D + AA D S π CA D 2 is nonsingular. Hence + A D S π CA D is also nonsingular and we conclude the first part of the proof. y applying Theorem 2.6 to this case, we get 3.5 and the proof is finished. Here we give two important consequences. Corollary 3.6. Let M be partitioned as in 1.3 and let S be as in 1.4. f A is nonsingular and rankm ranka + ranks D, then indm 1 if and only if A 2 + S π C is nonsingular. n this case [ M A R + 1 S π C A 1 S π ] CA 1 S π C S π CA 1 R 2 A 1 S π, where R is defined as in 2.8 with A D A 1. Corollary 3.7. Let M be partitioned as in 1.3. f rankm ranka D, then indm 1 if and only if W + A D CA D is nonsingular. n this case, M [ CA D A A D W 1] 2 A D. 3.6 Proof. From Theorem 2.6 we get [ A M D W 1 A A + D C W D W 1 2 C CA D ] A D, which can be rewritten as 3.6.

17 N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 Next, we explore the group invertibility of M when the block-rank condition and S π C hold. Further, we will derive a characterization in the case when S is nonsingular. Theorem 3.8. Let M be partitioned as in 1.3 and let S be as in 1.4. f S π C and rankm ranka D + ranks D, then indm 1 if and only if W + Z + CA D 2 SS is nonsingular, where Z S CA π S. n this case, where M R T [ + + A D S CA π S π C S π CA D S CA π A D + A D S CA D S CA D RT ] RT, 3.7 A D S S W 1, 3.8 A D W 1 S CA π + ZCA D A D W 1 SS + ZCA D 2 S CA π + ZCA D + ZCA D 2. 3.9 Proof. Let U 1 and V 1 be defined as in 2.1 and 2.5, respectively. Using AA π A π S CA π,s π C and SS π, applying Lemma 1.4 iii we get rankm 2 AA D S rank V 1 MMU 1 A 3 A D AA D S rank CA π S CAA D S 2 + CA π ranka 3 A D + rank S 2 + CA π + CA π S CA D 2 S ranka D + ranks 2 + CA π + CA π S CA D 2 S. Hence, on account that A π S π and by Lemma 1.4 i, it follows that rank M 2 rank A D + rank S D if and only if dim RS NS + CA π S + CA π S CA D 2 SS, which is equivalent to the fact that S + S π + S CA π S + CA D 2 SS is nonsingular. Hence, since S + S π is nonsingular, we conclude the first part of the proof. Let U and U 1 be defined as in 2.1. Using AA π A π S CA π and S π C, we get UMU 1 : X + Y, where A X 2 A D S CA 2 A D A π S CAA D A π S CAA D A Y π S π. C We note that XY and Y 2. y Lemma 1.3 ii, A D S S + S CA π + S CA D, S M U 1 X D + YX D 2 U. 3.1 To get a representation of X D we will apply Corollary 2.3. First, we observe that the Schur complement of A 2 A D in X is equal to WS. Moreover, we can easily see that WS D S W 1, WS π S π, indws 1.

N. Castro-González, M.F. Martínez-Serrano / Linear Algebra and its Applications 432 21 1691 172 171 Since inda 2 A D 1 also holds, applying Corollary 2.3 to the partitioned matrix X we conclude X D A D + A D 2 W 1 S CAA D A π S CA D A D 2 SS W 1 S W 1 S CAA D A π S CA D S W 1. 3.11 Using the notations 3.8 and 3.9, after some computations we get U 1 X D U RT and U 1 + A YU D S CAA π + A D S CA π S π C S CAA π S π CA D S CA π. Finally, by substituting the above expressions in 3.1 weget3.7. Corollary 3.9. Let M be partitioned as in 1.3 and let S be as in 1.4. f S is nonsigular and rankm ranka D + ranks, then indm 1 if and only if the matrix S 2 + CA π + CA D 2 S is nonsingular. n this case M is given by 3.7 with S S 1 and S π. We finish with a related result for the Drazin inverse. Theorem 3.1. Let M be partitioned as in 1.3 and let S be as in 1.4.frankM ranka D + ranks D, A D S and SCA D hold, then M D CA D A[WA D ] A 2 π A D S + S[ WS D ] 2 S CA π, where W AA D + A D CA D and W SS + S CA π S. Proof. We split M A 2 A D CAA D AA D CA D + AA π CA π A π S : X + Y. From S π CA π and SCA D we obtain CA π SS D C. Using this latter relation and A D S we get XY. Analogously we can see that YX. Then, M D X D + Y D. We can apply 1.7 to obtain the Drazin inverse of X, X D CA D A[WA D ] 2 A D, W AA D + A D CA D. On the other hand, the generalized Schur complement of S in Y, givenbyaa π A π S D CA π,isequal to zero. On account that A π S π and S π CA π, we can apply the symmetrical result of 1.7to obtain Y D A π S This completes the proof. [S WS D ] 2 S CA π, W SS + S CA π S. Acknowledgement The authors wish to thank to two anonymous referees for carefully reading this paper. References [1] J.K. aksalary, G.P.H. Styan, Generalized inverses of partitioned matrices in anachiewicz Schur form, Linear Algebra Appl. 354 22 41 47. [2] A. en-srael, T.N.E. Greville, Generalized nverses: Theory and Applications, second ed., Springer-Verlag, New York, 23. [3] J. enítez, N. Thome, The generalized Schur complement in group inverses and k + 1-potent matrices, Linear and Multilinear Algebra 54 6 26 45 413. [4] S.L. Campbell, C.D. Meyer Jr., Generalized nverses of Linear Transformations, Pitman, London, 1979 Dover, New York, 1991.

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