# Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

Authors: Xiaoji Liu; Hongwei Jin; Jelena Višnjić
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Representations of1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.


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## Body

## 1. Introduction

LetCm×n denote the set of all m×n complex matrices. We use R(A), N(A), and r(A) to denote the range, the null space, and the rank of a matrix A, respectively. The smallest nonnegative integer k such that r(Ak+1)=r(Ak) is the index of A which is denoted by ind(A). If ind(A)=r, the Drazin inverse of A is the unique matrix X∈Cm×m satisfying(1)XAX=X,AX=XA,Ak+1X=Ak,for all k≥r. The Drazin inverse of A is denoted by AD. In particular, when ind(A)=1, the matrix AD is called the group inverse of A and it is denoted by A#.Recall that the Moore-Penrose inverse of a matrixA∈Cm×n is a matrix X∈Cn×m which satisfies(2)1AXA=A,2XAX=X,3AX∗=AX,4XA∗=XA.The Moore-Penrose inverse of A is unique and it is denoted by A†.For anyA∈Cm×n, let Ai,j,…,k denote the set of all X∈Cn×m which satisfy equations i,j,…,k of (2). In this case X is a i,j,…,k-inverse of A, which we denote by A(i,j,…,k). Evidently, A1,2,3,4=A†.For a complex matrixM of the form(3)M=ABCD∈Cm+p×n+p,in the case when A is invertible, the Schur complement of A in M is defined by S=D-CA-1B. Similarly, if D is invertible, then the Schur complement of D in M is defined by G=A-BD-1C.It is well-known that ifA∈Cm×m is nonsingular then the invertibility of a matrix M is equivalent to the invertibility of the Schur complement of A in M and in that case the inverse of M is given by(4)M-1=A-1+A-1BS-1CA-1-A-1BS-1-S-1CA-1S-1.Expression (4) is called the Banachiewicz-Schur form of the matrix M. The notation of Schur complement was first introduced by Schur [1]. G. E. Trapp first defined the generalized Schur complement where the ordinary inverse was replaced by the generalized inverse. For a matrix M∈C(m+p)×(n+p) given by (3) and any fixed generalized inverse A-∈A1, the generalized Schur complement of A in M is defined by(5)SA-=D-CA-B.Similarly, for some fixedD-∈D1, the generalized Schur complement of D in M is defined by(6)GD-=A-BD-C.The Schur complements and generalized Schur complements were studied by a number of authors, which have applications in statistics, matrix theory, electrical network theory, discrete-time regulator problem, sophisticated techniques, and some other fields. It has been evident that the Schur complement plays an important role in many aspects of a matrix theory (see [2–8]).Let(7)X=A-+A-BS-CA--A-BS--S-CA-S-,where A-∈A1, S=D-CA-B, and S-∈S1, and let(8)Y=G--G-BD--D-CG-D-+D-CG-BD-,where D-∈D1, G=A-BD-C, and G-∈G1.For convenience, we will firstly state the following notations which are helpful in the proofs. We denote(9)Eα=I-α-α,Fα=I-αα-,απ=I-ααD,α∈A,S,D,G.In this paper, we will establish necessary and sufficient conditions under which matricesX and Y given by (7) and (8), respectively, belong to a given class of generalized inverses of the matrix M given by (3). Our interest focuses on the cases when 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix M are given by the Banachiewicz-Schur form. Also, we give representations of the Drazin inverse of 2×2 block matrix M under some conditions relating to the Schur complement of M, which generalized the results studied by Cvetković-Ilić [9]. Also, the quotient property and the first Sylvester identity based on the MP-inverse, group inverse, and Drazin inverse are given.We need some auxiliary lemmas.Lemma 1 (see [10]).
LetM, S, and X be defined by (3), (5), and (7), respectively. Then X∈M1,3 if and only if (10)A-∈A1,3,S-∈S1,3,FAB=0,FSC=0,the last two conditions being independent of the choice of A-∈A1 and S-∈S1 involved in FA and FS.Lemma 2 (see [10]).
LetM, S, and X be defined by (3), (5), and (7), respectively. Then X∈M1,4 if and only if (11)A-∈A1,4,S-∈S1,4,BES=0,CEA=0,the last two conditions being independent of the choice of A-∈A1 and S-∈S1 involved in FA and FS.Lemma 3 (see [10]).
LetM, S, and X be defined by (3), (5), and (7), respectively. Then X∈M2 if and only if (12)A-∈A1,2,S-∈S2.Lemma 4.
LetM, S, and X be defined by (3), (5), and (7), respectively. Then X∈M1,2,3 if and only if (13)A-∈A1,2,3,S-∈S1,2,3,FAB=0,FSC=0,the last two conditions being independent of the choice of A-∈A1 and S-∈S1 involved in FA and FS.Proof.
The conclusion follows by Lemmas1 and 2.Lemma 5.
LetM, S, and X be defined by (3), (5), and (7), respectively. Then X∈M1,2,4 if and only if(14)A-∈A1,2,4,S-∈S1,2,4,BES=0,CEA=0,the last two conditions being independent of the choice of A-∈A1 and S-∈S1 involved in EA and ES.Using the same method as in [10], we can prove the following.Lemma 6.
LetM, S, and X be defined by (3), (5), and (7), respectively. Then Y∈M1,2,3 if and only if (15)D-∈D1,2,3,G-∈G1,2,3,FGB=0,FDC=0,the last two conditions being independent of the choice of D-∈D1 and G-∈G1 involved in FD and FG.Lemma 7.
LetM, S, and X be defined by (3), (5), and (7), respectively. Then Y∈M1,2,4 if and only if (16)D-∈D1,2,4,G-∈G1,2,4,BED=0,CEG=0,the last two conditions being independent of the choice of D-∈D1 and G-∈G1 involved in FD and FG.Lemma 8 (see [11]).
LetM, S, and X be defined by (3), (5), and (7), respectively. Then X=M# if and only if (17)A-=A#,S-=S#,FAB=0,FSC=0,CEA=0,BES=0.


## 2. Representations of1,2,3-Inverse, 1,2,4-Inverse, and Group Inverse in terms of Banachiewicz-Schur Forms

Baksalary and Styan [10] presented the necessary and sufficient conditions under which 1, 2, 1,2, 1,3, and 1,4-inverses of block matrix M can be represented by the Banachiewicz-Schur form.Sheng and Chen [12] considered sufficient conditions under which M†, MD, and M# can be represented by both of the Banachiewicz-Schur forms at the same time.Cvetković-Ilić [9] gave necessary and sufficient conditions which ensure the representation of the MP-inverse of a block matrix M by both of the Banachiewicz-Schur forms.In this section, we will present necessary and sufficient conditions under which1,2,3, 1,2,4-inverses and the group inverse of a partitioned matrix M can be represented by the Banachiewicz-Schur form.For convenience, we first introduce the following notations:(18)N1i,j,k=X=A-+A-BS-CA--A-BS--S-CA-S-:A-∈Ai,j,k,S=D-CA-B,S-∈Si,j,k,N2i,j,k=Y=G--G-BD--D-CG-D-+D-CG-BD-:D-∈Di,j,k,G=A-BD-C,G-∈Gi,j,k.Theorem 9.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then the following happens.
(i) N 1 1,2 , 3 ⊆ N 2 1,2 , 3 ⊆ M 1,2 , 3 if and only if for arbitrary A-∈A1,2,3, D-∈D1,2,3 and G-∈G1,2,3, (19)FAB=0,FDC=0,FGB=0,and for any A-∈A1,2,3 and any S-∈S1,2,3, there exists D-∈D1,2,3 such that (20)FD=FS,EDS-=0,ESD-=0.
(ii)  N21,2,3⊆N11,2,3⊆M1,2,3 if and only if for arbitrary A-∈A1,2,3, D-∈D1,2,3 and S-∈S1,2,3, (21)FAB=0,FDC=0,FSC=0,and for any D-∈D1,2,3 and any G-∈G1,2,3, there exists A-∈A1,2,3 such that (22)FA=FG,EAG-=0,EGA-=0.
(iii)  N1{1,2,3}=N2{1,2,3}⊆M{1,2,3} if and only if for arbitrary A-∈A{1,2,3}, D-∈D{1,2,3}, (23)FAB=0,FDC=0,and for any A-∈A{1,2,3} and any S-∈S{1,2,3}, there exists D-∈D{1,2,3} such that FD=FS, EDS-=0, and ESD-=0, and for any D-∈D{1,2,3} and G-∈G{1,2,3}, there exists A-∈A{1,2,3} such that FA=FG, EAG-=0, and EGA-=0.Proof.
( i )   ( ⇐ : ) By Lemma 6, we have that arbitrary Y∈N21,2,3 belongs to M1,2,3. Now, we only need to prove that N11,2,3⊆N21,2,3. Take arbitrary(24)X1=A1-+A1-BS1-CA1--A1-BS1--S1-CA1-S1-∈N11,2,3.By assumption, for S1-, there exists 1,2,3-inverse of D which we denote by D1- such that FD=FS1, EDS1-=0, and ES1D1-=0. Let G=A-BD1-C. We will prove that G1-=A1-+A1-BS1-CA1-∈G1,2,3. By computation, we get(25)GG1-=A-BD1-CA1-+A1-BS1-CA1-=AA1-+AA1-BS1-CA1--BD1-CA1--BD1-CA1-BS1-CA1-=AA1-+BS1-CA1--BD1-CA1--BD1-DS1-CA1-+BD1-S1S1-CA1-=AA1-,which implies GG1-G=AA1-(A-BD1-C)=A-BD1-C=G. Also, (26)G1-GG1-=A1-+A1-BS1-CA1-AA1-=A1-+A1-BS1-CA1-=G1-,so, G1-∈G1,2,3. Since FD=FS1, EDS1-=0, and ES1D1-=0, a simple computation shows that(27)D1-CG1-=D1-CA1-+A1-BS1-CA1-=D1-CA1-+D1-CA1-BS1-CA1-=D1-CA1-+D1-D-S1S1-CA1-=D1-CA1-+D1-DS1-CA1--D1-S1S1-CA1-=S1-CA1-,G1-BD1-=A1-+A1-BS1-CA1-BD1-=A1-BD1-+A1-BS1-CA1-BD1-=A1-BD1-+A1-BS1-D-S1D1-=A1-BD1-+A1-BS1-DD1--A1-BS1-S1D1-=A1-BS1-,D1-+D1-CG1-BD1-=D1-+D1-CA1-+A1-BS1-CA1-BD1-=D1-+D1-CA1-BD1-+D1-CA1-BS1-CA1-BD1-=D1-+D1-D-S1D1-+D1-D-S1S1-D-S1D1-=S1-.Thus,(28)X1=A1-+A1-BS1-CA1--A1-BS1--S1-CA1-S1-=G1--G1-BD1--D1-CG1-D1-+D1-CG1-BD1-∈N2.Therefore, N1⊆N2⊆M1,2,3.
( ⇒ : ) Since N11,2,3⊆M1,2,3 and N21,2,3⊆M1,2,3, by Lemmas 4 and 6, we get that, for arbitrary A-∈A1,2,3, D-∈D1,2,3, and G-∈G1,2,3, the following hold: FAB=0, FGB=0, and FDC=0. Moreover, by N11,2,3⊆N21,2,3, it follows that, for any X∈N11,2,3, there exists some Y∈N21,2,3 such that X=Y. Hence, for any A-∈A1,2,3, S-∈S1,2,3, there exist D-∈D1,2,3 and G-∈G1,2,3 such that (29)A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-.A simple computation shows that (30)SS-=D-CA-BD-+D-CG-BD-=DD-+DD-CG-BD--CA-BD--CA-AG-BD-+CA-GG-BD-=DD-+CI-A-AG-BD-=DD-+CI-A-AA-BS-=DD-;thus, we get FS=FD. Since (31)D-+D-CG-BD-=D-+S-CA-BD-=D-+S-D-SD-=D-+S-DD--S-SD-=S-,we get D-=S-SD-; that is, ESD-=0. Similarly, (32)D-+D-CG-BD-=D-+D-CA-BS-=D-+D-D-SS-=D-+D-DS--D-SS-=S-,which implies EDS-=0.
( i i )   ( ⇐ : ) Similar to (i), since FAB=0 and FSC=0, we have N11,2,3⊆M1,2,3. Now, we will prove that N21,2,3⊆N11,2,3. Take arbitrary (33)Y1=G1--G1-BD1--D1-CG1-D1-+D1-CG1-BD1-∈N21,2,3.For G1-, there exists 1,2,3-inverse of A which we denote by A1- such that FA=FG1, EAG1-=0, and EG1A1-=0. Let S=D-CA1-B. A simple computation shows that (34)S1-=D1-+D1-CG1-BD1-∈S1,2,3,S1-CA1-=D1-CG1-,A1-BS1-=G1-BD1-,A1-+A1-BS1-CA1-=G1-.Hence,(35)Y1=G1--G1-BD1--D1-CG1-D1-+D1-CG1-BD1-=A1-+A1-BS1-CA1--A1-BS1--S1-CA1-S1-∈N11,2,3.Therefore, N21,2,3⊆N11,2,3⊆M1,2,3. The reverse part of the proof is similar to (i).
( i i i ) Combining (i) and (ii), we get (iii).The case for1,2,4-inverses is treated completely analogously and the corresponding result follows by taking adjoint.Theorem 10.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then the following happens.
(i)  N11,2,4⊆N21,2,4⊆M1,2,4 if and only if for arbitrary A-∈A1,2,4, D-∈D1,2,4, and G-∈G1,2,4, (36)CEA=0,BED=0,CEG=0,and for any A-∈A1,2,4 and any S-∈S1,2,4, there exists D-∈D1,2,4 such that (37)ES=ED,S-FD=0,D-FS=0.
(ii)  N21,2,4⊆N11,2,4⊆M1,2,4 if and only if for arbitrary A-∈A1,2,4, D-∈D1,2,4, and S-∈S1,2,4, (38)CEA=0,BED=0,BES=0,and for any D-∈D1,2,4 and any G-∈G1,2,4, there exists A-∈A1,2,4 such that (39)EA=EG,G-FA=0,A-FG=0.
(iii)  N11,2,4=N21,2,4⊆M1,2,4 if and only if for arbitrary A-∈A1,2,4, D-∈D1,2,4, (40)CEA=0,BED=0,and for any A-∈A1,2,4 and any S-∈S1,2,4, there exists D-∈D1,2,4 such that ES=ED, S-FD=0, and D-FS=0, and for any D-∈D1,2,4 and any G-∈G1,2,4, there exists A-∈A1,2,4 such that EA=EG, G-FA=0, and A-FG=0.Similarly, we get the following results.Theorem 11.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then the following happens.
(i) For arbitrary X∈N11,2,3 given by (7), there exists Y∈N21,2,3 such that X=Y∈M1,2,3 if and only if A-∈A1,2,3 and S-∈S1,2,3 and there exist D-∈D1,2,3 and G-∈G1,2,3 such that (41)FAB=0,FDC=0,FGB=0,FD=FS,EDS-=0,ESD-=0.
(ii) For arbitrary Y∈N21,2,3 given by (8), there exists X∈N11,2,3 such that X=Y∈M1,2,3 if and only if D-∈D1,2,3 and G-∈G1,2,3 and there exist A-∈A1,2,3 and S-∈S1,2,3 such that (42)FAB=0,FDC=0,FSC=0,FA=FG,EAG-=0,EGA-=0.Theorem 12.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then the following happens.
(i) For arbitrary X∈N11,2,4 given by (7), there exists Y∈N21,2,4 such that X=Y∈M1,2,4 if and only if A-∈A1,2,4 and S-∈S1,2,4 and there exist D-∈D1,2,4 and G-∈G1,2,4 such that (43)CEA=0,BED=0,CEG=0,ES=ED,S-FD=0,D-FS=0.
(ii) For arbitrary Y∈N21,2,4 given by (8), there exists X∈N11,2,4 such that X=Y∈M1,2,4 if and only if D-∈D1,2,4 and G-∈G1,2,4 and there exist A-∈A1,2,4 and S-∈S1,2,4 such that (44)CEA=0,BED=0,BES=0,EA=EG,G-FA=0,A-FG=0.Theorem 13.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then(45)M†=A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-if and only if(46)A-=A†,D-=D†,S-=S†,G-=G†,FAB=0,FDC=0,CEA=0,BED=0,ED=ES,FD=FS.Proof.
By Theorems11 and 12, we get(47)M†=A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-if and only if(48)A-=A†,D-=D†,S-=S†,G-=G†,EDS-=0,ESD-=0,EAG-=0,EGA-=0,D-FS=0,S-FD=0,G-FA=0,A-FG=0,FAB=0,FDC=0,CEA=0,BED=0,ES=ED,EA=EG,FD=FS,FA=FG.Since ES=ED and EDD-=0, then ESD-=0. Thus, (49)ES=ED,EA=EG,FD=FS,FA=FGimplies(50)EDS-=0,ESD-=0,EAG-=0,EGA-=0,D-FS=0,S-FD=0,G-FA=0,A-FG=0.Now, we only need to prove that (51)FAB=0,FDC=0,CEA=0,BED=0,ES=ED,EA=EG,FD=FS,FA=FGis equivalent to (46). Denote G′=A-+A-BS-CA-. By FAB=0, BED=0, and FS=FD, a simple computation shows that (52)GG′=A-BD-CA-+A-BS-CA-=AA-+AA-BS-CA--BD-CA--BD-DS-CA-+BD-SS-CA-=AA-.Similarly, by FDC=0, CEA=0, and ES=ED, we get G′G=A-A. Moreover, GG′G=G and G′GG′=G′. Thus, G′=G† and EA=EG and FA=FG. Therefore, (51) is equivalent to (46).Corollary 14 (see [9]).
LetM, S, and G be defined by (3), (5), and (6), respectively. Then(53)M†=A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-if and only if (54)A-=A†,D-=D†,S-=S†,G-=G†,FAB=0,FDC=0,FSC=0,CEA=0,BED=0,BES=0.Proof.
By Corollary14 from [9], it is easy to conclude that (54) is equivalent to (46).Corollary 15.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then(55)M†=A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-if and only if (56)A-=A†,D-=D†,S-=S†,G-=G†,FAB=0,FDC=0,CEA=0,BED=0,EA=EG,FA=FG.Corollary 16.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then(57)M†=A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-if and only if(58)A-=A†,D-=D†,S-=S†,G-=G†,FAB=0,FDC=0,FGB=0,CEA=0,BED=0,CEG=0.Theorem 17.
LetM, S, and G be defined by (3), (5), and (6), respectively. Then(59)M#=A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-if and only if one of the following conditions holds.
(i)  (60) A - = A # , D - = D # , S - = S # , G - = G # , F A B = 0 , F D C = 0 , F S C = 0 , C E A = 0 , B E D = 0 , B E S = 0 .
(ii)  (61) A - = A # , D - = D # , S - = S # , G - = G # , F A B = 0 , F D C = 0 , F G B = 0 , C E A = 0 , B E D = 0 , C E G = 0 .Proof.
( i ) By Lemma 8,(62)M#=A-+A-BS-CA--A-BS--S-CA-S-if and only if(63)A-=A#,S-=S#,FAB=0,FSC=0,CEA=0,BES=0.Similarly,(64)M#=G--G-BD--D-CG-D-+D-CG-BD-if and only if(65)D-=D#,G-=G#,FGB=0,FDC=0,CEG=0,BED=0.Since the group inverse is unique if it exists, combining (63) and (65), we get(66)M#=A-+A-BS-CA--A-BS--S-CA-S-=G--G-BD--D-CG-D-+D-CG-BD-if and only if(67)A-=A#,D-=D#,S-=S#,G-=G#,FAB=0,FGB=0,FSC=0,FDC=0,BES=0,CEA=0,BED=0,CEG=0.
Now, we only need to prove (67) is equivalent to (60). Denote G′=A-+A-BS-CA-. If (60) holds, then(68)GG′=A-BD-CA-+A-BS-CA-=AA-+AA-BS-CA--BD-CA--BD-CA-BS-CA-=AA-+BS-CA--BD-CA--BD-D-SS-CA-=AA-,G′G=A-+A-BS-CA-A-BD-C=A-A-A-BD-C-A-BS-CA-A-A-BS-CA-BD-C=A-A-A-BD-C-A-BS-C-A-BS-D-SD-C=A-A.Now, it is easy to get (69)GG′G=AA-A-BD-C=A-BD-C=G,G′GG′=A-AA-+A-BS-CA-=A-+A-BS-CA-=G′and GG′=G′G. Thus, G′=G#. Hence, G#G=A-A and GG#=AA-. Now, we get FAB=FGB=0 and CEA=CEG=0, which means (60) implies (67). Obviously, (67) implies (60). Thus, (67) is equivalent to (60).
( i i ) The proof is similar to the proof of (i).


## 3. Representations of Drazin Inverse in terms of Banachiewicz-Schur Forms

The Drazin inverse of a square matrix has various applications in singular differential equations and singular difference equations, Markov chains, and iterative methods. Actually, in 1979 Campbell and Meyer [13] posed the problem of finding an explicit representation for the Drazin inverse of a complex block matrix M, in terms of its blocks. No formula for MD has yet been offered without any restrictions upon the blocks. Many papers have considered this open problem and each of them offered a formula for the Drazin inverse and specific conditions for the 2×2 block matrix M (see [9, 11, 12, 14–22]).Wei [23] proved that MD can be expressed by AD+ADBSDCAD-ADBSD-SDCADSD if AπC=0, BAπ=0, CSπ=0, SπB=0, and DSπ=0, where Aπ=I-AAD and Sπ=I-SSD.Cvetković-Ilić [9] generalized the result from [23] and gave the representation of MD under simpler conditions than in [23].Sheng and Chen [12] proved that, under the conditions AπB=0, BDπ=0, CAπ=0, DπC=0, and Sπ=Dπ=0, MD can be represented by both of Banachiewicz-Schur forms.In the following, we will present conditions more general than those in [9, 12, 23] which ensure the representation of MD by the Banachiewicz-Schur form.Theorem 18.
LetM be defined by (3) and S=D-CADB be the generalized Schur complement of A in M. If(70)ADBSπ=0,SDCAπ=0,AπBSD=0,SπCAD=0,then(71)MD=AD+ADBSDCAD-ADBSD-SDCADSD.Proof.
Denote the right side of (71) by X. By simple computations, we show that(72)MX=AAD-I-AADBSDCADI-AADBSDI-SSDCADSSD,XM=ADA-ADBSDCI-ADAADBI-SDSSDCI-ADASDS.Since ADBSπ=0, SDCAπ=0, AπBSD=0, and SπCAD=0, (72) reduce to (73)MX=AAD00SSD,XM=ADA00SDS.Obviously, MX=XM. On the other hand, (74)XMX=ADA00SDSAD+ADBSDCAD-ADBSD-SDCADSD=AD+ADBSDCAD-ADBSD-SDCADSD=X.Now, we need to prove that M-M2N is a nilpotent matrix. Since (75)M-M2N=A-I-AADBSDCI-ADAI-AADBI-SDSI-SSDCI-ADASI-SSD,by ADBSπ=0, SDCAπ=0, we get (76)M-M2N=AI-ADA00SI-SSD,which is nilpotent and ind(M-M2N)⩾max⁡{ind(A),ind(S)}.Corollary 19 (see [9]).
LetM be given by (3) and S=D-CADB be the generalized Schur complement of A in M. If (77)ADBSπ=0,SDCAπ=0,AπBSD=0,SπCAD=0and if there exists a nonnegative integer k such that MkP=0, then(78)MD=AD+ADBSDCAD-ADBSD-SDCADSD,where P=Aπ00Sπ.Corollary 20 (see [23]).
LetM be given by (3) and S=D-CADB be the generalized Schur complement of A in M. If (79)AπB=0,BSπ=0,CAπ=0,SπC=0,DSπ=0,then(80)MD=AD+ADBSDCAD-ADBSD-SDCADSD.Theorem 21.
LetM be defined as in (3) and G=A-BDDC be the generalized Schur complement of D in M. If (81)DDCGπ=0,GDBDπ=0,GπBDD=0,DπCGD=0,then(82)MD=GD-GDBDD-DDCGDDD+DDCGDBDD.Proof.
The proof is analogous to that of Theorem18.Theorem 22.
LetM be defined by (3) and S=D-CADB, G=A-BDDC be the generalized Schur complement of A in M and D in M, respectively. If any of the conditions hold(i)
AπB=0, CAπ=0, BDπ=0, DπC=0, SπCAD=0, ADBSπ=0,(ii)
AπB=0, CAπ=0, BDπ=0, DπC=0, GπBDD=0, DDCGπ=0,then(83)MD=AD+ADBSDCAD-ADBSD-SDCADSD=GD-GDBDD-DDCGDDD+DDCGDBDD.Proof.
( i ) Denote X=AD+ADBSDCAD. Firstly, we prove GD=AD+ADBSDCAD. It is easy to see (84)GX=A-BDDCAD+ADBSDCAD=AAD+AADBSDCAD-BDDCAD-BDDCADBSDCAD=AAD+AADBSDCAD-BDDCAD-BDDD-SSDCAD=AAD+AADBSDCAD-BDDCAD-BDDDSDCAD+BDDSSDCAD=AADby AπB=0, BDπ=0 and SπCAD=0, and(85)XG=AD+ADBSDCADA-BDDC=ADA+ADBDDC-ADBSDCADA-ADBSDCADBDDC=ADA+ADBDDC-ADBSDCADA-ADBSDD-SDDC=ADA+ADBDDC-ADBSDCADA-ADBSDDDDC+ADBSDSDDC=ADAby CAπ=0, DπC=0, and ADBSπ=0. Thus, GX=XG. Moreover, (86)XGX=ADAAD+ADBSDCAD=AD+ADBSDCAD=X,GkXG=GkADA=Gk-1A-BDDCADA=Gk-1A-BDDC=Gk,by using CAπ=0. Therefore, GGD=AAD. Since AπB=0 and CAπ=0, we get GπB=0, CGπ=0. Now, we can deduce that(87)MD=AD+ADBSDCAD-ADBSD-SDCADSD,by ADBSπ=0, CAπ=0, AπB=0, and SπCAD=0 and (88)MD=GD-GDBDD-DDCGDDD+DDCGDBDD,by CGπ=0, BDπ=0, GπB=0, and DπC=0. Then, (89)MD=AD+ADBSDCAD-ADBSD-SDCADSD=GD-GDBDD-DDCGDDD+DDCGDBDDfollows by the uniqueness of the Drazin inverse.
( i i ) Similarly, we denote Y=DD+DDCGDBDD. In the same way, we can easily get SY=DDD by CAπ=0, BDπ=0, DπCGD=0, and YS=DDD by AπB=0, CGπ=0, and GπBDD=0. Since (90)YSY=DDDDD+DDCGDBDD=DD+DDCGDBDD=Y,SkYS=Sk-1D-CADBDDD=Sk-1D-CADB=Sk,by using BDπ=0, we obtain SD=DD+DDCGDBDD. Thus, we have SSD=DDD which implies BSπ=0 and SπC=0, where we used BDπ=0 and DπC=0. Now, by BSπ=0, SπC=0, AπB=0, and CAπ=0, we get (91)MD=AD+ADBSDCAD-ADBSDSDCADSD,and by DπC=0, BDπ=0, GπBDD=0, and DDCGπ=0, we have (92)MD=GD-GDBDD-DDCGDDD+DDCGDBDD.Therefore,(93)MD=AD+ADBSDCAD-ADBSD-SDCADSD=GD-GDBDD-DDCGDDD+DDCGDBDD.Corollary 23 (see [12]).
LetM be defined by (3) and S=D-CADB, G=A-BDDC be the generalized Schur complement of A in M and D in M, respectively. If(94)AπB=0,BDπ=0,CAπ=0,DπC=0,Sπ=Dπ,then(95)MD=AD+ADBSDCAD-ADBSD-SDCADSD=GD-GDBDD-DDCGDDD+DDCGDBDD.Proof.
According to the proof of Theorem22(ii), we have SD=Y=DD+DDCGDBDD which implies SSD=DDD by SY=DDD. Now, it is sufficient to conclude that (95) holds.


## 4. Quotient Property and the First Sylvester Identity in terms of the Generalized Schur Complement

Crabtree and Haynsworth in [18] showed a quotient formula for Schur complement of a matrix. After that, the formula was extended by Ostrowski in 1971 [24] and in 1973 [19].Let(96)Q=ABECDFGHJ,M=ABCD,M2=BEDF,M3=DFHJ,(97)M4=CDGH.If Q and M are square and nonsingular, then(98)QM=Q/AM/A=AEGJA-ABGHAMA-1AECFA.This is the so-called quotient property which is proven by Crabtree and Haynsworth in [18].The first Sylvester identity is that(99)QM3=Q/DM3/D=MD-M2DM3D-1M4D,where M3 and D are nonsingular.Sheng and Chen [12] studied the quotient properties and the first Sylvester identity with respect to MP, group, and Drazin inverses. Cvetković-Ilić in [9] derived the properties under conditions different than those used in [12].In this section, we will give the quotient properties and the first Sylvester identity of MP, group, and Drazin inverses under the conditions which are much simpler than those of [9, 12]. Notice that the proofs of the following results are almost identical to the proofs of the corresponding results from [12]. We will state the theorems without proofs.We first give the quotient properties based on MP, group, and Drazin inverses. For convenience, we denote(100)MAp=D-CA†B,MAg=D-CA#B,MAd=D-CADB,MDp=A-BD†C,MDg=A-BD#C,MDd=A-BDDC.Theorem 24.
LetQ=ABECDFGHJ and M=ABCD. If M satisfies the conditions of Corollary 15, then (101)QMp=Q/ApM/App.Theorem 25.
LetQ=ABECDFGHJ and M=ABCD. If M satisfies the conditions of Theorem 11(i) or (ii), then(102)QMg=Q/AgM/Agg.Theorem 26.
LetQ=ABECDFGHJ and M=ABCD. If M satisfies the conditions of Theorem 18, then (103)QMd=Q/AdM/Add.The first Sylvester identity with respect to MP, group, and Drazin inverses is expressed as follows.Theorem 27.
LetQ, M, M2, M3, and M4 be the forms of (97). If M3 satisfies the conditions of Corollary 15, then (104)QM3p=Q/DpM3/Dpp=MDp-M2DpM3Dp†M4Dp.Theorem 28.
LetQ, M, M2, M3, and M4 be the forms of (97). If M3 satisfies the conditions of Theorem 11(i) or (ii), then (105)QM3g=Q/DgM3/Dgg=MDg-M2DgM3Dg#M4Dg.Theorem 29.
LetQ, M, M2, M3, and M4 be the forms of (97). If M3 satisfies the conditions of Theorem 18, then(106)QM3d=Q/DdM3/Ddd=MDd-M2DdM3DdDM4Dd.


## 5. Applications of the Generalized Schur Complement

The Schur complements and generalized Schur complements were shown playing an important role in various applied mathematical settings, such as statistics, matrix theory, electrical network theory, discrete-time regulator problem, sophisticated techniques, and many other fields.In this section, we will give an application of the generalized Schur complement in the solution of a linear system. Using the generalized Schur complement, we can split a larger system into two small linear systems by the following steps.Let(107)Mx=ybe a linear system. Applying the block Gaussian elimination to the system, we have (108)AB0D-CA†Bx1x2=y1y2-CA†y1.Hence, we get (109)Ax1+Bx2=y1;Sx2=y2-CA†y1;that is, (110)Ax1=y1-Bx2,Sx2=y2-CA†y1.Now, the solution of system (107) can be obtained by the two small linear systems above. In this case, the operation can be significantly simplified. We will also notice that the Moore-Penrose inverse of A can be replaced by other generalized inverses, such as the group inverse, the Drazin inverse, and the generalized inverse of A or even the ordinary inverse A-1.In the following, we will give the group inverse solutions of the linear system.Theorem 30.
Let(111)Mx=ybe a linear system. Suppose M satisfies all the conditions of Theorem 17, partitioning x and y as(112)x=x1x2,y=y1y2,which have appropriate sizes with M. If y∈R(M), then the solution x=M#y of linear system (111) can be expressed as (113)x1=A#y1-Bx2,x2=S#y2-CA#y1,where S=D-CA#B.Proof.
Sincey∈R(M), we conclude that x=M#y is the solution of linear system (111). By Theorem 17, we can get the following: (114)x=M#y=A#+A#BS#CA#-A#BS#-S#CA#S#y1y2=A#y1+A#BS#CA#y1-A#BS#y2S#y2-CA#y1=x1x2.Now, it is easy to see the solution x=M#y can be expressed as (115)x1=A#y1-Bx2,x2=S#y2-CA#y1,which is also the group inverse solutions of the two small linear systems of (110), respectively.



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*Source: 9236281-2016-10-18.xml*