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Further results on the group inverses and Drazin inverses of anti-triangular block matrices
Applied Mathematics and Computation 218 (2012) 8978–8986
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Further results on the group inverses and Drazin inverses of anti-triangular block matrices Xifu Liu a,b, Hu Yang a,⇑ a b
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China School of Basic Science, East China Jiaotong University, Nanchang 330013, China
a r t i c l e
i n f o
a b s t r a c t In this paper, the sufficient and necessary conditions for the existence of the group inverse for a class of 2 2 anti-triangular block matrices M with some certain conditions satisfied, and representations of the group inverse of M are obtained, which can be regarded as the extension of Bu et al. [C. Bu, J. Zhao, J. Zheng, Group inverse for a class 2 2 block matrices over skew fields, Appl. Math. Comput. 204 (2008) 45–49] and [C. Bu, J. Zhao, K. Zhang, Some results on group inverses of block matrices over skew fields, Electron. J. Linear Algebra, 18 (2009) 117–125]. Further, we use these results to determine the expressions of the Drazin inverse of M. Finally, two numerical examples are given to illustrate our results. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Drazin inverse Group inverse Block matrix Idempotent matrix
1. Introduction Let Cmn be the set of complex m n matrices. The symbols r(A) and A(1) will denote the rank and any {1}-inverse of a given matrix A 2 C mn . Further, AD will stand for the Drazin inverse of A, i.e., the unique matrix satisfying the equations [1]
Ak XA ¼ Ak ;
XAX ¼ X;
AX ¼ XA;
ð1:1Þ
where k = Ind(A), the index of A, is the smallest nonnegative integer k such that
rðAkþ1 Þ ¼ rðAk Þ: If IndðAÞ ¼ 1, then the Drazin inverse of A is reduced to the group inverse, denote by A# . If IndðAÞ ¼ 0, then AD ¼ A1 . In addition, we denote Ap ¼ I AAD (or Ap ¼ I AA# ), especially, if A is idempotent, then Ap ¼ I A. The Drazin inverse is very useful, and has various applications in singular differential or difference equations, Markov chains, cryptography, iterative method and numerical analysis. In 1979, Campbell and Meyer first proposed an open problem A B of finding a formula for the Drazin inverse of M ¼ in terms of the blocks of the partition over complex fields, where C D A and D are square matrices but need not to be the same size. To the best of our knowledge, up to now, this problem has not been solved completely. However, some special cases of the Drazin inverse and group inverse of M were studied under certain conditions [3,7,10,11,15,17–19,21,23], and the expressions of the Drazin inverse and group inverse in terms of Banachiewicz–Schur forms were also investigated in [12–14,16,20]. Specially, when D ¼ 0,
⇑ Corresponding author. E-mail address: [email protected] (H. Yang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.02.058
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
M¼
A
B
C
0
8979
ð1:2Þ
is the so-called anti-triangular block matrix, where A 2 C nn ; B 2 C nm and C 2 C mn . In recent years, the Drazin inverse and group inverse for M formed by (1.2) have been considered by many authors. For instance, Deng [6] established some expressions of the Drazin inverse of M when BCAp ¼ 0 and CAD B is either invertible or equal to zero. Deng and Wei [9] got explicit expressions for the Drazin inverse of the anti-triangular block matrix M under the following different cases
ðiÞ ABC ¼ 0; ðiiÞ BCAp ¼ 0;
ðI Ap ÞBC ¼ 0; ðiiiÞ Ap AB ¼ 0;
BCðI Ap Þ ¼ 0:
In [22], Castro–González and Dopazo provided the representations for MD under two different conditions
ðiÞ A ¼ I;
B ¼ I;
ðiiÞ CAAD ¼ C;
AD BC ¼ BCAD :
Bu et al. [2,4] determined the conditions for the existence of M # in the case B ¼ A or C ¼ A, and presented the representations of M# . For the matrix
M¼
A ; 0
A C
if A2 ¼ A, in [2], the authors showed that M # exists if and only if rðCÞ ¼ rðCACÞ; for each matrix A, need not to be idempotent, if rðCÞ P rðAÞ, in [4], they showed that M# exists if and only if rðCÞ ¼ rðAÞ ¼ rðCAÞ ¼ rðACÞ. In recent paper [8], Bu et al. considered the group inverse of M in the case A can be expressed as the linear combination or product combination of matrices B AX þ YB A and C. Cao and Li [25] mainly considered the group inverse of matrix with the form over skew fields. Zhou B 0 et al. [26] established the conditions for the existence and the representations for the group inverse of block matrix M with one or two full rank sub-blocks. In the next section, some improved results on the group inverse of the anti-triangular block matrix M are obtained, which extend the results given by Bu et al. [2,4]. Moreover, applying our new results, several representations of the Drazin inverse of anti-triangular block matrix M are derived under certain conditions. Before giving the main results, we first introduce some important lemmas as follows. Lemma 1.1 [24]. Let A 2 C mn ; B 2 C mk ; C 2 C ln and D 2 C lk . Then
r ð A B Þ ¼ rðAÞ þ rððI AAð1Þ ÞBÞ ¼ rððI BBð1Þ ÞAÞ þ rðBÞ; r
A
r
¼ rðAÞ þ rðCðI Að1Þ AÞÞ ¼ rðAðI C ð1Þ CÞÞ þ rðCÞ;
C A
B
C
D
¼ rðAÞ þ r
0
ðI AAð1Þ ÞB
CðI Að1Þ AÞ
D CAð1Þ B
ð1:3Þ ð1:4Þ
! ;
ð1:5Þ
where the Að1Þ is an arbitrary {1}-inverses of A. Lemma 1.2. Let B 2 C mn ; C 2 C nm . If rðBÞ ¼ rðBCBÞ, then ðBCÞ# and ðCBÞ# exist. Furthermore, ðBCÞ# ¼ BððCBÞ# Þ2 C. Proof. In view of rðBÞ ¼ rðBCBÞ, we have
rðBÞ ¼ rðBCBÞ 6 rðBCÞ 6 rðBÞ; which means
rðBÞ ¼ rðBCÞ: Similarly, we can deduce
rðBÞ ¼ rðCBÞ: Hence
rðBÞ ¼ rðBCBÞ ¼ rðBCÞ ¼ rðCBÞ: And the condition r(B) = r(BC) shows that there exists a matrix X e Cmn such that B = BCX. Therefore
rðBCÞ ¼ rðBCBÞ ¼ rðBCBCXÞ 6 rðBCÞ2 :
ð1:6Þ
On the other hand,
rðBCÞ P rðBCÞ2 :
ð1:7Þ
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X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
Combining (1.6) and (1.7) gives r(BC) = r(BC)2. Therefore, ðBCÞ# exists.The existence of ðCBÞ# can be proved similarly, we omit the details. With the fact ðBCÞD ¼ BððCBÞD Þ2 C, then ðBCÞ# ¼ BððCBÞ# Þ2 C is evident. h Lemma 1.3 [2]. Let M ¼
B B ¼ rðAÞ. , where A 2 C mm and B 2 C nm . Then M # exists if and only if A# exists and r A A
0 0
Lemma 1.4 [5]. Let P; Q 2 C mm , and IndðPÞ ¼ r; IndðQ Þ ¼ s, such that PQ ¼ 0. Then
ðP þ Q ÞD ¼ Q p
s1 X
Q i ðPD Þiþ1 þ
r1 X ðQ D Þiþ1 Pi Pp :
i¼0
i¼0
Specially, (i) If P2 ¼ 0, then
ðP þ Q ÞD ¼ Q D þ ðQ D Þ2 P: (ii) If Q 2 ¼ 0, then
ðP þ Q ÞD ¼ PD þ Q ðPD Þ2 : 2. Main results In this section, we first establish the necessary and sufficient conditions for the existence of M # under some weaker conditions than those in [2,4], where M is given by (1.2), then present the explicit expressions for M # . Furthermore, we use our new results to derive some representations for the Drazin inverse of M under certain assumptions. First, we consider the case that A is idempotent. CA ¼ C, then Theorem 2.1. Let M be partitioned as (1.2). If A2¼ pA and A B (i) M# exists if and only if ðCBÞ# exists and r ¼ rðCBÞ, where Ap ¼ I A. CB (ii) If M# exists, then
0 B M# ¼ @
A ABðCBÞ# C 2BðCBÞp C
BðCBÞp þ Ap BððCBÞ# Þ2
Ap BððCBÞ# Þ2 C
þBðCBÞ#
ðCBÞp C þ ðCBÞ# C
ðCBÞ#
1 C A;
ð2:1Þ
where ðCBÞp ¼ I CBðCBÞ# . Proof. (i) Recall that, M # exists if and only if rðMÞ ¼ rðM 2 Þ. Because A is idempotent, A is a {1}-inverses of A. Then, from (1.5), we have
rðMÞ ¼ r
2
rðM Þ ¼ r
A B C 0
¼ rðAÞ þ r
A þ BC
AB
C
CB
¼r
0 Ap B 0
CB
! ð2:2Þ
;
A AB BCB C
CB
¼r
A
BCB
C 0 !2 1 ! 0 Ap B A 0 Ap BCB ¼ rðAÞ þ r @ : ¼ rðAÞ þ r 2 0 ðCBÞ 0 CB
¼ rðAÞ þ r
0
0
Ap BCB
CAp
ðCBÞ2
!
ð2:3Þ
According to(2.2) and (2.3), we know that M # exists if and only if p A B ¼ rðCBÞ. ðCBÞ# exists and r CB
0 0
Ap B CB
#
exists, by Lemma 1.3, which is equivalent to
(ii) Let N be the right hand side of (2.1). We can compute that,
NM ¼
A þ Ap BðCBÞ# C BðCBÞp C
ABðCBÞp Ap BðCBÞ#
ðCBÞp C
CBðCBÞ#
!
;
ð2:4Þ
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
0 B MN ¼ @
1 ABðCBÞp Ap BðCBÞ# C A:
A þ Ap BðCBÞ# C 2ABðCBÞp C þBðCBÞp C ðCBÞp C
If M # exists, then ðCBÞ# exists and r
Ap BðCBÞp ¼ 0;
ð2:5Þ
CBðCBÞ#
p A B ¼ rðCBÞ, by (1.4), we can see CB
or ABðCBÞp ¼ BðCBÞp :
ð2:6Þ
Substituting (2.6) into (2.5) yields that
NM ¼
8981
A þ Ap BðCBÞ# C BðCBÞp C
BðCBÞp Ap BðCBÞ#
ðCBÞp C
CBðCBÞ#
! ¼ MN:
Moreover,
0
A ABðCBÞ# C 2ABðCBÞp C
ABðCBÞp þ ABðCBÞ# þ Ap BðCBÞ#
1
Ap BððCBÞ# Þ2 C
þAp BððCBÞ# Þ2
C C A
ðCBÞp C þ ðCBÞ# C
ðCBÞ# 1 BðCBÞp þ BðCBÞ# C þAp BððCBÞ# Þ2 C A ¼ N;
B NMN ¼ B @ 0
A ABðCBÞ# C 2BðCBÞp C
B ¼B @
Ap BððCBÞ# Þ2 C ðCBÞp C þ ðCBÞ# C
ðCBÞ#
and, it is easy to verify that MNM ¼ M. Hence, N ¼ M# . Then we finish the proof of this theorem. h The following result is a special case of Theorem 2.1, which can be found in [2]. A B Corollary 2.1. Let M ¼ , where A; B 2 C nn , such that A2 ¼ A. Then A 0 (i) M# exists if and only if rðBÞ ¼ rðBABÞ. (ii) If M # exists, then #
M ¼
ðABÞp A þ AðBAÞ# ðBAÞ#
ðBAÞ# B þ BððABÞ# Þ2 ðABÞ#
ðABÞp A þ ðABÞ# A
ðABÞ#
ðABÞp A þ ðABÞ# A ðBAÞ# ðABÞp A þ ðABÞ# A
¼
!
! ðBAÞ# B þ ðBAÞ# ðABÞ# AB ðABÞ# ; ðABÞ#
where ðABÞp ¼ I ABðABÞ# . p A B ¼ rðABÞ, which are equivalent to the facts AB # that ðABÞ exists and rðBÞ ¼ rðABÞ. Next, we prove that these conditions are equivalent to rðBÞ ¼ rðBABÞ. If rðBÞ ¼ rðBABÞ, according to Lemma 1.2, ðABÞ# exists and rðBÞ ¼ rðABÞ. On the other hand, if ðABÞ# exists and rðBÞ ¼ rðABÞ, then Proof (i). In view of Theorem 2.1, M # exists if and only if ðABÞ# exists and r
rðBÞ ¼ rðABÞ ¼ rðABABÞ 6 rðBABÞ 6 rðBÞ: Hence, rðBÞ ¼ rðBABÞ. The proof of part (i) is complete. (ii) The expression of M # can be got directly from Theorem 2.1 h Similarly, we can prove the counterpart of Theorem 2.1. Theorem 2.2. Let M be partitioned as (1.2). If A2 ¼ A and AB ¼ B, then (i) M # exists if and only if ðCBÞ# exists and r CAp CB ¼ rðCBÞ, where Ap ¼ I A. (ii) If M # exists, then
0 B M# ¼ @
A BðCBÞ# CA 2BðCBÞp C BððCBÞ# Þ2 CAp p
ðCBÞ C þ ððCBÞ# Þ2 CAp þ ðCBÞ# C
1 BðCBÞp þ BðCBÞ# C A; ðCBÞ#
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X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
where ðCBÞp ¼ I CBðCBÞ# . As an application of Theorem 2.1 or Theorem 2.2, we can deduce following result. Corollary 2.2. Let M be partitioned as (1.2). If A2 ¼ A; AB ¼ B and CA ¼ C, then (i) M # exists if and only if ðCBÞ# exists. (ii) If M # exists, then
A BðCBÞ# C 2BðCBÞp C ðCBÞp C þ ðCBÞ# C
M# ¼
! BðCBÞp þ BðCBÞ# ; ðCBÞ#
where ðCBÞp ¼ I CBðCBÞ# . The following result is a special case of Theorem 2.2, which can be found in [2]. Corollary 2.3. Let M ¼
A C
A , where A; C 2 C nn , such that A2 ¼ A. Then 0
(i) M# exists if and only if rðCÞ ¼ rðCACÞ. (ii) If M # exists, then #
M ¼
AðCAÞp þ AðCAÞ# AððCAÞ# Þ2 C
AðCAÞp þ AðCAÞ#
ðCAÞ# C ðCAÞ# þ ððCAÞ# Þ2 C
ðCAÞ#
!
AðCAÞp þ AðCAÞ# ðACÞ# ðCAÞ# C ðCAÞ# þ ðCAÞ# ðACÞ# AC
¼
! AðCAÞp þ AðCAÞ# ; ðCAÞ#
where ðCBÞp ¼ I CBðCBÞ# . In particular, let A = I, then Corollary 2.3 reduces to following result, which can be found in [22]. Corollary 2.4. Let M ¼
I C
I . Then 0
(i) M # exists if and only if C # exists. (ii) If M # exists, then
M# ¼
Cp
Cp þ C#
C# C
C #
! :
Next, we present a representation of MD. Theorem 2.3. Let M be partitioned as (1.2). If A2 ¼ A; CAp B ¼ 0 and ðCBÞ# exists, then
0
A ABðCBÞ# CA 2ABðCBÞp CA þ ABðCBÞp CAp
B B ABððCBÞ# Þ2 CAp Ap BððCBÞ# Þ2 CAp B M ¼B B þAp BðCBÞp CA Ap BððCBÞ# Þ2 C Ap BððCBÞ# Þ3 CAp @ D
ðCBÞp CA þ ðCBÞ# C þ ððCBÞ# Þ2 CAp
0 B B B ¼B B @
AðBCÞD Ap þ Ap BðCBÞp CA Ap ðBCÞD Ap Ap ðBCÞD Ap ððBCÞD Þ2 Ap ðCBÞp CA þ ðCBÞ# C þ ððCBÞ# Þ2 CAp
ABðCBÞ þ BðCBÞ C C C þAp BððCBÞ# Þ2 C C A ð2:7Þ
1
ABðCBÞp þ BðCBÞ# C C C þAp BððCBÞ# Þ2 C: C A ðCBÞ#
A C
AB 0
and Q ¼
MD ¼ PD þ Q ðP D Þ2 : D
#
ðCBÞ#
A ABðCBÞ# CA 2ABðCBÞp CA þ ABðCBÞp CAp
Proof. Consider the splitting M ¼ P þ Q , where P ¼ Lemma 1.4, we obtain
1
p
Next, we calculate P . Rewrite P as the sum P ¼ P1 þ P2 , where P1 ¼ P1 P2 ¼ 0. By Lemma 1.4, it follows that
0 CAp
0 0
0 0
Ap B . Note that Q 2 ¼ 0 and PQ ¼ 0, applying 0
and P 2 ¼
ð2:8Þ
A CA
AB . We see that P 21 ¼ 0 and 0
PD ¼ PD2 þ ðPD2 Þ2 P1 : p
ð2:9Þ #
#
#
Since the condition CA B ¼ 0 is equivalent to CB ¼ CAB, then ðCABÞ exists if and only if ðCBÞ exists, and ðCABÞ ¼ ðCBÞ# . According to the assumptions and Corollary 2.2, we know that P# 2 exists, and
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
A ABðCBÞ# CA 2ABðCBÞp CA ABðCBÞp þ ABðCBÞ#
P# 2 ¼
ðCBÞp CA þ ðCBÞ# CA
8983
! :
ðCBÞ#
ð2:10Þ
Substituting (2.10) into (2.9) gives
0
1 ABðCBÞp þ ABðCBÞ# C A:
A ABðCBÞ# CA 2ABðCBÞp CA
B PD ¼ @ þABðCBÞp CAp ABððCBÞ# Þ2 CAp p
p
# 2
#
ðCBÞ CA þ ðCBÞ C þ ððCBÞ Þ CA Substituting (2.11) into (2.8) gives (2.7).
ðCBÞ
ð2:11Þ
#
h
Note that the condition CA ¼ C or AB ¼ B implies CAp B ¼ 0, then we can deduce the following results. Corollary 2.5. Let M be partitioned as (1.2). If A2 ¼ A; CA ¼ C and ðCBÞ# exists, then
0
A ABðCBÞ# C 2ABðCBÞp C
ABðCBÞp þ BðCBÞ#
1
þAp BðCBÞp C Ap ðBCÞD
þAp BððCBÞ# Þ2
C C: A
ðCBÞp C þ ðCBÞ# C
ðCBÞ#
B MD ¼ B @
Corollary 2.6. Let M be partitioned as (1.2). If A2 ¼ A; AB ¼ B and ðCBÞ# exists, then
0 B M ¼@
1 BðCBÞp þ BðCBÞ# C A:
A BðCBÞ# CA 2BðCBÞp CA
D
p
p
D
p
þBðCBÞ CA ðBCÞ A
ðCBÞp CA þ ðCBÞ# C þ ððCBÞ# Þ2 CAp
ðCBÞ#
Theorem 2.4. Let M be partitioned as (1.2), such that rðCÞ 6 rðBÞ. If AC ð1Þ C ¼ A, then (i) M# exists if and only if rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ. (ii) If M # exists, then #
M ¼
ðBCÞp AðBCÞ#
ðBCÞ# B ðBCÞp AðBCÞ# AðBCÞ# B
CðBCÞ#
CðBCÞ# AðBCÞ# B
! ð2:12Þ
;
where ðBCÞp ¼ I BCðBCÞ# . Proof. (i) Recall that, M # exists if and only if rðMÞ ¼ rðM2 Þ. Since AC ð1Þ C ¼ A, we can show that
rðMÞ ¼ r
A B C
2
rðM Þ ¼ r
! ¼r
0
0
B
C
0
A2 þ BC
AB
CA
CB
! ¼ rðCÞ þ rðBÞ;
!
¼r
BC
0
CA
CB
:
Therefore, it is obvious that the condition is sufficient. Now we show that condition is also necessary.
rðMÞ ¼ rðM 2 Þ ) rðCÞ þ rðBÞ ¼ r
BC
0
CA CB
6 rðBCÞ þ rð CA CB Þ 6 rðBCÞ þ rðCÞ 6 rðCÞ þ rðBÞ;
hence, rðBCÞ þ rðCÞ ¼ rðBÞ þ rðCÞ, which implies rðBÞ ¼ rðBCÞ. On the other hand,
rðCÞ þ rðBÞ ¼ r
BC CA
0 CB
6r
BC CA
þ rðCBÞ ¼ r
BC ð1Þ
CAC C
þ rðCBÞ 6 rðCÞ þ rðCBÞ 6 rðCÞ þ rðBÞ;
namely, rðCBÞ þ rðCÞ ¼ rðBÞ þ rðCÞ, means rðBÞ ¼ rðCBÞ. Therefore, rðBÞ ¼ rðCBÞ ¼ rðBCÞ, and rðBÞ 6 rðCÞ. Combining rðCÞ 6 rðBÞ shows that
rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ: (ii) From rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ, we know that ðBCÞ# and ðCBÞ# exist. Moreover, the following equalities hold
AðBCÞp ¼ 0;
CðBCÞp ¼ 0 and ðBCÞp B ¼ 0:
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X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
Let the right hand side of (2.12) be N. A simple computation shows that
MN ¼
BCðBCÞ#
ðBCÞp AðBCÞ# B
0
CðBCÞ# B
! ¼ NM:
Furthermore, we can deduce that
MNM ¼ M
and NMN ¼ N:
According to the definition of group inverse, we know that N ¼ M # .
h
Remark that the condition AC ð1Þ C ¼ A in Theorem 2.4 can be replaced by RðA Þ # RðC Þ. And rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ is equivalent to rðBÞ ¼ rðCÞ ¼ rðCBCÞ ¼ rðBCBÞ. The following result is a special case of Theorem 2.4, which can be found in [4]. A B Corollary 2.7. Let M ¼ , where A; B 2 C nn , such that rðAÞ 6 rðBÞ. Then A 0 (i) M # exists if and only if rðBÞ ¼ rðAÞ ¼ rðBAÞ ¼ rðABÞ. (ii) If M# exists, then #
M ¼
ðBAÞp AðBAÞ#
ðBAÞ# B ðBAÞp AðBAÞ# AðBAÞ# B
AðBAÞ#
AðBAÞ# AðBAÞ# B
! ;
where ðBAÞp ¼ I BAðBAÞ# . Similarly, we can derive the following theorem, which can be regarded as a same case with the Theorem 1.2 in [25] given by Cao and Li, where they presented the necessary and sufficient conditions to the existence and the representation of the group AX A inverse for block matrix . B 0
Theorem 2.5. Let M be partitioned as (1.2), such that rðBÞ 6 rðCÞ. If BBð1Þ A ¼ A, or equivalently RðAÞ # RðBÞ, then
(i) M# exists if and only if rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ. (ii) If M# exists, then
M ¼
!
ðBCÞ# AðBCÞp
ðBCÞ# B
CðBCÞ# CðBCÞ# AðBCÞ# AðBCÞp
CðBCÞ# AðBCÞ# B
#
;
where ðBCÞp ¼ I BCðBCÞ# . Corollary 2.8. Let M ¼
A C
A , where A; C 2 C nn , such that rðAÞ 6 rðCÞ. Then 0
(i) M# exists if and only if rðCÞ ¼ rðAÞ ¼ rðCAÞ ¼ rðACÞ. (ii) If M # exists, then #
M ¼
!
ðACÞ# AðACÞp
ðACÞ# A
CðACÞ# CðACÞ# AðACÞ# AðACÞp
CðACÞ# AðACÞ# A
;
where ðACÞp ¼ I ACðACÞ# . Based on Theorem 2.4 and Theorem 2.5, we can deduce the following results. Theorem 2.6. Let M be partitioned as (1.2). If AC ð1Þ C ¼ A, or equivalently RðA Þ # RðC Þ, and rðCÞ ¼ rðCBCÞ, then IndðMÞ 6 2, and
MD ¼
ðBCÞp AðBCÞ#
ðBCÞ# B ðBCÞp AðBCÞ# AðBCÞ# B
CðBCÞ#
CðBCÞ# AðBCÞ# B
where ðBCÞp ¼ I BCðBCÞ# .
! :
ð2:13Þ
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
8985
Proof. Consider the splitting
M¼P¼
A
C ð1Þ CB
C
0
!
0 B C ð1Þ CB
þ
0
! D
¼ P þ Q:
0
We observe that Q 2 ¼ 0 and PQ ¼ 0. Applying Lemma 1.4, we obtain
MD ¼ PD þ Q ðPD Þ2 :
ð2:14Þ
D
Now, we calculate P . Since rðCÞ ¼ rðCBCÞ, according to Lemma 1.2, it follows that rðCÞ ¼ rðBCÞ ¼ rðCBÞ ¼ rðCBCÞ, so P satisfies the conditions in Theorem 2.4 (i), that is to say P# exists. Moreover, it is clearly that C ð1Þ CBCðBCÞ# ¼ C ð1Þ C. Hence, by Theorem 2.4, we have
! C ð1Þ CðBCÞ# B F C AðBCÞ# AðBCÞ# B ; CðBCÞ# AðBCÞ# B
F C AðBCÞ# CðBCÞ#
#
P ¼
ð2:15Þ
where F C ¼ I C ð1Þ C. Substituting (2.15) into (2.14}) gives (2.13). It is easy to verify that
MM p ¼ MðI MMD Þ ¼
0 ðBCÞp B 0
!
0
;
which means that M 2 M p ¼ 0, i.e., IndðMÞ 6 2. h Theorem 2.7. Let M be partitioned as (1.2). If BBð1Þ A ¼ A, or equivalently RðAÞ # RðBÞ, and rðBÞ ¼ rðBCBÞ, then IndðMÞ 6 2, and
M ¼
!
ðBCÞ# AðBCÞp
ðBCÞ# B
CðBCÞ# CðBCÞ# AðBCÞ# AðBCÞp
CðBCÞ# AðBCÞ# B
D
;
where ðBCÞp ¼ I BCðBCÞ# . Remark. Although the expressions of M # in Theorem 2.4 (or Theorem 2.5) and M D in Theorem 2.6 (or Theorem 2.7) have the same form, we observe that their needful conditions are different, because the condition rðBÞ ¼ rðCÞ in Theorem 2.6 (or Theorem 2.7) is not satisfied.
3. Examples In this section, we give two examples to illustrate Theorem 2.3 and Theorem 2.7. A B Example 3.1. Consider the block matrix M ¼ , where C 0
0
1 0 1
B C A ¼ @ 0 1 0 A; 0
0
0
1
1
1
B B¼@ 1
0
1
C 1 A; 1
1
C¼
1 1 1
2 0 2
:
It is easy to verify that A2 ¼ A; CAp B ¼ 0 and ðCBÞ# exists. Furthermore, we can calculate that
0
1 0 0 1 B C A ¼ @ 0 0 0 A; 0 0 1 p
ðCBÞ# ¼
1 1 ; 0 0
ðCBÞp ¼
0 1 : 0 1
Hence, applying Theorem 2.3, we get
1 6 1 6 2 2 C B 1 C B 3 0 3 1 C B D B M ¼B 7 1 7 2 2 C C: C B 1 1 1 1 A @ 1 0
2
0
2
0
0
Example 3.2. Consider the block matrix M ¼
A C
B , where 0
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X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
0
1 0 1
B C A ¼ @ 0 2 1 A; 0
0
0
1
1 1
1
B C B ¼ @ 0 1 A;
C¼
0 0
0
1
1 0
1 1 1
:
We observe that RðAÞ # RðBÞ, and rðBÞ ¼ rðBCBÞ. Furthermore, we can compute that
0 ðBCÞ# ¼
2 4 3
1B @2 4 0
0 0
1
C 1 A;
0 ðBCÞp ¼
0
0 0
1
1
1B C @ 0 0 1 A: 2 0 0 2
By Theorem 2.7, we have
1 0 0 3 2 2 C B 3 2 2 C B0 0 C 1B D 0 0 0 0 0 C M ¼ B C: 4B C B 4 4 1 2 6 A @ 0 4 4 4 4 0
Acknowledgements The authors would like to thank the editors and the reviewers for their valuable comments and helpful suggestions, which improved the paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11171361) and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20110191110033). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, second ed., Springer, New York, 2003. C. Bu, J. Zhao, J. Zheng, Group inverse for a class 2 2 block matrices over skew fields, Appl. Math. Comput. 204 (2008) 45–49. X. Chen, R.E. Hartwig, The group inverse of a triangular matrix, Linear Algebra Appl. 237–238 (1996) 97–108. C. Bu, J. Zhao, K. Zhang, Some results on group inverses of block matrices over skew fields, Electron. J. Linear Algebra 18 (2009) 117–125. R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207–217. C. Deng, Generalized Drazin inverses of anti-triangular block matrices, J. Math. Anal. Appl. 368 (2010) 1–8. C. Deng, Dragana S. Cvetkovic´-Ilic´, Y. Wei, Some results on the Drazin inverse of operator matrices, Linear Multilinear Algebra 58 (2010) 503–521. C. Bu, K. Zhang, J. Zhao, Some results on the group inverse of the block matrix with a sub-block of linear combination or product combination of matrices over skew fields, Linear Multilinear Algebra 58 (2010) 957–966. C. Deng, Y. Wei, A note on the Drazin inverse of an anti-triangular matrix, Linear Algebra Appl. 431 (2009) 1910–1922. L. Guo, X. Du, Representations for the Drazin inverses of 2 2 block matrices, Appl. Math. Comput. 217 (2010) 2833–2842. E. Dopazo, M.F. Martínez-Serrano, Further results on the representation of the Drazin inverse of a 2 2 block matrix, Linear Algebra Appl. 432 (2010) 1896–1904. N. Castro-González, M.F. Martínez-Serrano, Drazin inverse of partitioned matrices in terms of Banachiewicz–Schur forms, Linear Algebra Appl. 432 (2010) 1691–1702. M.F. Martínez-Serrano, N. Castro-González, On the Drazin inverse of block matrices and generalized Schur complement, Appl. Math. Comput. 215 (2009) 2733–2740. Dragana S. Cvetkovic´-Ilic´, Expression of the Drazin and MP-inverse of partitioned matrix and quotient identity of generalized Schur complement, Appl. Math. Comput. 213 (2009) 18–24. Dragana S. Cvetkovic´-Ilic´, J. Chen, Z. Xu, Explicit representations of the Drazin inverse of block matrix and modified matrix, Linear Multilinear Algebra 57 (2009) 355–364. C. Deng, A note on the Drazin inverses with Banachiewicz–Schur forms, Appl. Math. Comput. 213 (2009) 230–234. Dragana S. Cvetkovic´-Ilic´, A note on the representation for the Drazin inverse of 2 2 block matrices, Linear Algebra Appl. 429 (2008) 242–248. N. Castro-González, E. Dopazo, M.F. Martínez-Serrano, On the Drazin inverse of the sum of two operators and its application to operator matrices, J. Math. Anal. Appl. 350 (2008) 207–215. X. Li, Y. Wei, A note on the representations for the Drazin inverse of 2 2 block matrices, Linear Algebra Appl. 423 (2007) 332–338. R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverses of a 2 2 block matrix, SIAM. J. Matrix Anal. Appl. 27 (2006) 757–771. N. Castro-González, E. Dopazo, J. Robles, Formulas for the Drazin inverse of special block matrices, Appl. Math. Comput. 174 (2006) 252–270. N. Castro-González, E. Dopazo, Representations of the Drazin inverse for a class of block matrices, Linear Algebra Appl. 400 (2005) 253–269. Y. Wei, Expressions for the Drazin inverse of a 2 2 block matrix, Linear Multilinear Algebra 45 (1998) 131–146. G. Marsaglia, G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra 2 (1974) 269–292. C. Cao, J. Li, A note on the group inverse of some 2 2 block matrices over skew fields, Appl. Math. Comput. 217 (2011) 10271–10277. J. Zhou, C. Bu, Y. Wei, Group inverse for block matrices and some related sign analysis, Linear and Multilinear Algebra, doi:10.1080/ 03081087.2011.625498 (2011).
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Further results on the group inverses and Drazin inverses of anti-triangular block matrices Xifu Liu a,b, Hu Yang a,⇑ a b
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China School of Basic Science, East China Jiaotong University, Nanchang 330013, China
a r t i c l e
i n f o
a b s t r a c t In this paper, the sufficient and necessary conditions for the existence of the group inverse for a class of 2 2 anti-triangular block matrices M with some certain conditions satisfied, and representations of the group inverse of M are obtained, which can be regarded as the extension of Bu et al. [C. Bu, J. Zhao, J. Zheng, Group inverse for a class 2 2 block matrices over skew fields, Appl. Math. Comput. 204 (2008) 45–49] and [C. Bu, J. Zhao, K. Zhang, Some results on group inverses of block matrices over skew fields, Electron. J. Linear Algebra, 18 (2009) 117–125]. Further, we use these results to determine the expressions of the Drazin inverse of M. Finally, two numerical examples are given to illustrate our results. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Drazin inverse Group inverse Block matrix Idempotent matrix
1. Introduction Let Cmn be the set of complex m n matrices. The symbols r(A) and A(1) will denote the rank and any {1}-inverse of a given matrix A 2 C mn . Further, AD will stand for the Drazin inverse of A, i.e., the unique matrix satisfying the equations [1]
Ak XA ¼ Ak ;
XAX ¼ X;
AX ¼ XA;
ð1:1Þ
where k = Ind(A), the index of A, is the smallest nonnegative integer k such that
rðAkþ1 Þ ¼ rðAk Þ: If IndðAÞ ¼ 1, then the Drazin inverse of A is reduced to the group inverse, denote by A# . If IndðAÞ ¼ 0, then AD ¼ A1 . In addition, we denote Ap ¼ I AAD (or Ap ¼ I AA# ), especially, if A is idempotent, then Ap ¼ I A. The Drazin inverse is very useful, and has various applications in singular differential or difference equations, Markov chains, cryptography, iterative method and numerical analysis. In 1979, Campbell and Meyer first proposed an open problem A B of finding a formula for the Drazin inverse of M ¼ in terms of the blocks of the partition over complex fields, where C D A and D are square matrices but need not to be the same size. To the best of our knowledge, up to now, this problem has not been solved completely. However, some special cases of the Drazin inverse and group inverse of M were studied under certain conditions [3,7,10,11,15,17–19,21,23], and the expressions of the Drazin inverse and group inverse in terms of Banachiewicz–Schur forms were also investigated in [12–14,16,20]. Specially, when D ¼ 0,
⇑ Corresponding author. E-mail address: [email protected] (H. Yang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.02.058
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
M¼
A
B
C
0
8979
ð1:2Þ
is the so-called anti-triangular block matrix, where A 2 C nn ; B 2 C nm and C 2 C mn . In recent years, the Drazin inverse and group inverse for M formed by (1.2) have been considered by many authors. For instance, Deng [6] established some expressions of the Drazin inverse of M when BCAp ¼ 0 and CAD B is either invertible or equal to zero. Deng and Wei [9] got explicit expressions for the Drazin inverse of the anti-triangular block matrix M under the following different cases
ðiÞ ABC ¼ 0; ðiiÞ BCAp ¼ 0;
ðI Ap ÞBC ¼ 0; ðiiiÞ Ap AB ¼ 0;
BCðI Ap Þ ¼ 0:
In [22], Castro–González and Dopazo provided the representations for MD under two different conditions
ðiÞ A ¼ I;
B ¼ I;
ðiiÞ CAAD ¼ C;
AD BC ¼ BCAD :
Bu et al. [2,4] determined the conditions for the existence of M # in the case B ¼ A or C ¼ A, and presented the representations of M# . For the matrix
M¼
A ; 0
A C
if A2 ¼ A, in [2], the authors showed that M # exists if and only if rðCÞ ¼ rðCACÞ; for each matrix A, need not to be idempotent, if rðCÞ P rðAÞ, in [4], they showed that M# exists if and only if rðCÞ ¼ rðAÞ ¼ rðCAÞ ¼ rðACÞ. In recent paper [8], Bu et al. considered the group inverse of M in the case A can be expressed as the linear combination or product combination of matrices B AX þ YB A and C. Cao and Li [25] mainly considered the group inverse of matrix with the form over skew fields. Zhou B 0 et al. [26] established the conditions for the existence and the representations for the group inverse of block matrix M with one or two full rank sub-blocks. In the next section, some improved results on the group inverse of the anti-triangular block matrix M are obtained, which extend the results given by Bu et al. [2,4]. Moreover, applying our new results, several representations of the Drazin inverse of anti-triangular block matrix M are derived under certain conditions. Before giving the main results, we first introduce some important lemmas as follows. Lemma 1.1 [24]. Let A 2 C mn ; B 2 C mk ; C 2 C ln and D 2 C lk . Then
r ð A B Þ ¼ rðAÞ þ rððI AAð1Þ ÞBÞ ¼ rððI BBð1Þ ÞAÞ þ rðBÞ; r
A
r
¼ rðAÞ þ rðCðI Að1Þ AÞÞ ¼ rðAðI C ð1Þ CÞÞ þ rðCÞ;
C A
B
C
D
¼ rðAÞ þ r
0
ðI AAð1Þ ÞB
CðI Að1Þ AÞ
D CAð1Þ B
ð1:3Þ ð1:4Þ
! ;
ð1:5Þ
where the Að1Þ is an arbitrary {1}-inverses of A. Lemma 1.2. Let B 2 C mn ; C 2 C nm . If rðBÞ ¼ rðBCBÞ, then ðBCÞ# and ðCBÞ# exist. Furthermore, ðBCÞ# ¼ BððCBÞ# Þ2 C. Proof. In view of rðBÞ ¼ rðBCBÞ, we have
rðBÞ ¼ rðBCBÞ 6 rðBCÞ 6 rðBÞ; which means
rðBÞ ¼ rðBCÞ: Similarly, we can deduce
rðBÞ ¼ rðCBÞ: Hence
rðBÞ ¼ rðBCBÞ ¼ rðBCÞ ¼ rðCBÞ: And the condition r(B) = r(BC) shows that there exists a matrix X e Cmn such that B = BCX. Therefore
rðBCÞ ¼ rðBCBÞ ¼ rðBCBCXÞ 6 rðBCÞ2 :
ð1:6Þ
On the other hand,
rðBCÞ P rðBCÞ2 :
ð1:7Þ
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X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
Combining (1.6) and (1.7) gives r(BC) = r(BC)2. Therefore, ðBCÞ# exists.The existence of ðCBÞ# can be proved similarly, we omit the details. With the fact ðBCÞD ¼ BððCBÞD Þ2 C, then ðBCÞ# ¼ BððCBÞ# Þ2 C is evident. h Lemma 1.3 [2]. Let M ¼
B B ¼ rðAÞ. , where A 2 C mm and B 2 C nm . Then M # exists if and only if A# exists and r A A
0 0
Lemma 1.4 [5]. Let P; Q 2 C mm , and IndðPÞ ¼ r; IndðQ Þ ¼ s, such that PQ ¼ 0. Then
ðP þ Q ÞD ¼ Q p
s1 X
Q i ðPD Þiþ1 þ
r1 X ðQ D Þiþ1 Pi Pp :
i¼0
i¼0
Specially, (i) If P2 ¼ 0, then
ðP þ Q ÞD ¼ Q D þ ðQ D Þ2 P: (ii) If Q 2 ¼ 0, then
ðP þ Q ÞD ¼ PD þ Q ðPD Þ2 : 2. Main results In this section, we first establish the necessary and sufficient conditions for the existence of M # under some weaker conditions than those in [2,4], where M is given by (1.2), then present the explicit expressions for M # . Furthermore, we use our new results to derive some representations for the Drazin inverse of M under certain assumptions. First, we consider the case that A is idempotent. CA ¼ C, then Theorem 2.1. Let M be partitioned as (1.2). If A2¼ pA and A B (i) M# exists if and only if ðCBÞ# exists and r ¼ rðCBÞ, where Ap ¼ I A. CB (ii) If M# exists, then
0 B M# ¼ @
A ABðCBÞ# C 2BðCBÞp C
BðCBÞp þ Ap BððCBÞ# Þ2
Ap BððCBÞ# Þ2 C
þBðCBÞ#
ðCBÞp C þ ðCBÞ# C
ðCBÞ#
1 C A;
ð2:1Þ
where ðCBÞp ¼ I CBðCBÞ# . Proof. (i) Recall that, M # exists if and only if rðMÞ ¼ rðM 2 Þ. Because A is idempotent, A is a {1}-inverses of A. Then, from (1.5), we have
rðMÞ ¼ r
2
rðM Þ ¼ r
A B C 0
¼ rðAÞ þ r
A þ BC
AB
C
CB
¼r
0 Ap B 0
CB
! ð2:2Þ
;
A AB BCB C
CB
¼r
A
BCB
C 0 !2 1 ! 0 Ap B A 0 Ap BCB ¼ rðAÞ þ r @ : ¼ rðAÞ þ r 2 0 ðCBÞ 0 CB
¼ rðAÞ þ r
0
0
Ap BCB
CAp
ðCBÞ2
!
ð2:3Þ
According to(2.2) and (2.3), we know that M # exists if and only if p A B ¼ rðCBÞ. ðCBÞ# exists and r CB
0 0
Ap B CB
#
exists, by Lemma 1.3, which is equivalent to
(ii) Let N be the right hand side of (2.1). We can compute that,
NM ¼
A þ Ap BðCBÞ# C BðCBÞp C
ABðCBÞp Ap BðCBÞ#
ðCBÞp C
CBðCBÞ#
!
;
ð2:4Þ
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
0 B MN ¼ @
1 ABðCBÞp Ap BðCBÞ# C A:
A þ Ap BðCBÞ# C 2ABðCBÞp C þBðCBÞp C ðCBÞp C
If M # exists, then ðCBÞ# exists and r
Ap BðCBÞp ¼ 0;
ð2:5Þ
CBðCBÞ#
p A B ¼ rðCBÞ, by (1.4), we can see CB
or ABðCBÞp ¼ BðCBÞp :
ð2:6Þ
Substituting (2.6) into (2.5) yields that
NM ¼
8981
A þ Ap BðCBÞ# C BðCBÞp C
BðCBÞp Ap BðCBÞ#
ðCBÞp C
CBðCBÞ#
! ¼ MN:
Moreover,
0
A ABðCBÞ# C 2ABðCBÞp C
ABðCBÞp þ ABðCBÞ# þ Ap BðCBÞ#
1
Ap BððCBÞ# Þ2 C
þAp BððCBÞ# Þ2
C C A
ðCBÞp C þ ðCBÞ# C
ðCBÞ# 1 BðCBÞp þ BðCBÞ# C þAp BððCBÞ# Þ2 C A ¼ N;
B NMN ¼ B @ 0
A ABðCBÞ# C 2BðCBÞp C
B ¼B @
Ap BððCBÞ# Þ2 C ðCBÞp C þ ðCBÞ# C
ðCBÞ#
and, it is easy to verify that MNM ¼ M. Hence, N ¼ M# . Then we finish the proof of this theorem. h The following result is a special case of Theorem 2.1, which can be found in [2]. A B Corollary 2.1. Let M ¼ , where A; B 2 C nn , such that A2 ¼ A. Then A 0 (i) M# exists if and only if rðBÞ ¼ rðBABÞ. (ii) If M # exists, then #
M ¼
ðABÞp A þ AðBAÞ# ðBAÞ#
ðBAÞ# B þ BððABÞ# Þ2 ðABÞ#
ðABÞp A þ ðABÞ# A
ðABÞ#
ðABÞp A þ ðABÞ# A ðBAÞ# ðABÞp A þ ðABÞ# A
¼
!
! ðBAÞ# B þ ðBAÞ# ðABÞ# AB ðABÞ# ; ðABÞ#
where ðABÞp ¼ I ABðABÞ# . p A B ¼ rðABÞ, which are equivalent to the facts AB # that ðABÞ exists and rðBÞ ¼ rðABÞ. Next, we prove that these conditions are equivalent to rðBÞ ¼ rðBABÞ. If rðBÞ ¼ rðBABÞ, according to Lemma 1.2, ðABÞ# exists and rðBÞ ¼ rðABÞ. On the other hand, if ðABÞ# exists and rðBÞ ¼ rðABÞ, then Proof (i). In view of Theorem 2.1, M # exists if and only if ðABÞ# exists and r
rðBÞ ¼ rðABÞ ¼ rðABABÞ 6 rðBABÞ 6 rðBÞ: Hence, rðBÞ ¼ rðBABÞ. The proof of part (i) is complete. (ii) The expression of M # can be got directly from Theorem 2.1 h Similarly, we can prove the counterpart of Theorem 2.1. Theorem 2.2. Let M be partitioned as (1.2). If A2 ¼ A and AB ¼ B, then (i) M # exists if and only if ðCBÞ# exists and r CAp CB ¼ rðCBÞ, where Ap ¼ I A. (ii) If M # exists, then
0 B M# ¼ @
A BðCBÞ# CA 2BðCBÞp C BððCBÞ# Þ2 CAp p
ðCBÞ C þ ððCBÞ# Þ2 CAp þ ðCBÞ# C
1 BðCBÞp þ BðCBÞ# C A; ðCBÞ#
8982
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
where ðCBÞp ¼ I CBðCBÞ# . As an application of Theorem 2.1 or Theorem 2.2, we can deduce following result. Corollary 2.2. Let M be partitioned as (1.2). If A2 ¼ A; AB ¼ B and CA ¼ C, then (i) M # exists if and only if ðCBÞ# exists. (ii) If M # exists, then
A BðCBÞ# C 2BðCBÞp C ðCBÞp C þ ðCBÞ# C
M# ¼
! BðCBÞp þ BðCBÞ# ; ðCBÞ#
where ðCBÞp ¼ I CBðCBÞ# . The following result is a special case of Theorem 2.2, which can be found in [2]. Corollary 2.3. Let M ¼
A C
A , where A; C 2 C nn , such that A2 ¼ A. Then 0
(i) M# exists if and only if rðCÞ ¼ rðCACÞ. (ii) If M # exists, then #
M ¼
AðCAÞp þ AðCAÞ# AððCAÞ# Þ2 C
AðCAÞp þ AðCAÞ#
ðCAÞ# C ðCAÞ# þ ððCAÞ# Þ2 C
ðCAÞ#
!
AðCAÞp þ AðCAÞ# ðACÞ# ðCAÞ# C ðCAÞ# þ ðCAÞ# ðACÞ# AC
¼
! AðCAÞp þ AðCAÞ# ; ðCAÞ#
where ðCBÞp ¼ I CBðCBÞ# . In particular, let A = I, then Corollary 2.3 reduces to following result, which can be found in [22]. Corollary 2.4. Let M ¼
I C
I . Then 0
(i) M # exists if and only if C # exists. (ii) If M # exists, then
M# ¼
Cp
Cp þ C#
C# C
C #
! :
Next, we present a representation of MD. Theorem 2.3. Let M be partitioned as (1.2). If A2 ¼ A; CAp B ¼ 0 and ðCBÞ# exists, then
0
A ABðCBÞ# CA 2ABðCBÞp CA þ ABðCBÞp CAp
B B ABððCBÞ# Þ2 CAp Ap BððCBÞ# Þ2 CAp B M ¼B B þAp BðCBÞp CA Ap BððCBÞ# Þ2 C Ap BððCBÞ# Þ3 CAp @ D
ðCBÞp CA þ ðCBÞ# C þ ððCBÞ# Þ2 CAp
0 B B B ¼B B @
AðBCÞD Ap þ Ap BðCBÞp CA Ap ðBCÞD Ap Ap ðBCÞD Ap ððBCÞD Þ2 Ap ðCBÞp CA þ ðCBÞ# C þ ððCBÞ# Þ2 CAp
ABðCBÞ þ BðCBÞ C C C þAp BððCBÞ# Þ2 C C A ð2:7Þ
1
ABðCBÞp þ BðCBÞ# C C C þAp BððCBÞ# Þ2 C: C A ðCBÞ#
A C
AB 0
and Q ¼
MD ¼ PD þ Q ðP D Þ2 : D
#
ðCBÞ#
A ABðCBÞ# CA 2ABðCBÞp CA þ ABðCBÞp CAp
Proof. Consider the splitting M ¼ P þ Q , where P ¼ Lemma 1.4, we obtain
1
p
Next, we calculate P . Rewrite P as the sum P ¼ P1 þ P2 , where P1 ¼ P1 P2 ¼ 0. By Lemma 1.4, it follows that
0 CAp
0 0
0 0
Ap B . Note that Q 2 ¼ 0 and PQ ¼ 0, applying 0
and P 2 ¼
ð2:8Þ
A CA
AB . We see that P 21 ¼ 0 and 0
PD ¼ PD2 þ ðPD2 Þ2 P1 : p
ð2:9Þ #
#
#
Since the condition CA B ¼ 0 is equivalent to CB ¼ CAB, then ðCABÞ exists if and only if ðCBÞ exists, and ðCABÞ ¼ ðCBÞ# . According to the assumptions and Corollary 2.2, we know that P# 2 exists, and
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
A ABðCBÞ# CA 2ABðCBÞp CA ABðCBÞp þ ABðCBÞ#
P# 2 ¼
ðCBÞp CA þ ðCBÞ# CA
8983
! :
ðCBÞ#
ð2:10Þ
Substituting (2.10) into (2.9) gives
0
1 ABðCBÞp þ ABðCBÞ# C A:
A ABðCBÞ# CA 2ABðCBÞp CA
B PD ¼ @ þABðCBÞp CAp ABððCBÞ# Þ2 CAp p
p
# 2
#
ðCBÞ CA þ ðCBÞ C þ ððCBÞ Þ CA Substituting (2.11) into (2.8) gives (2.7).
ðCBÞ
ð2:11Þ
#
h
Note that the condition CA ¼ C or AB ¼ B implies CAp B ¼ 0, then we can deduce the following results. Corollary 2.5. Let M be partitioned as (1.2). If A2 ¼ A; CA ¼ C and ðCBÞ# exists, then
0
A ABðCBÞ# C 2ABðCBÞp C
ABðCBÞp þ BðCBÞ#
1
þAp BðCBÞp C Ap ðBCÞD
þAp BððCBÞ# Þ2
C C: A
ðCBÞp C þ ðCBÞ# C
ðCBÞ#
B MD ¼ B @
Corollary 2.6. Let M be partitioned as (1.2). If A2 ¼ A; AB ¼ B and ðCBÞ# exists, then
0 B M ¼@
1 BðCBÞp þ BðCBÞ# C A:
A BðCBÞ# CA 2BðCBÞp CA
D
p
p
D
p
þBðCBÞ CA ðBCÞ A
ðCBÞp CA þ ðCBÞ# C þ ððCBÞ# Þ2 CAp
ðCBÞ#
Theorem 2.4. Let M be partitioned as (1.2), such that rðCÞ 6 rðBÞ. If AC ð1Þ C ¼ A, then (i) M# exists if and only if rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ. (ii) If M # exists, then #
M ¼
ðBCÞp AðBCÞ#
ðBCÞ# B ðBCÞp AðBCÞ# AðBCÞ# B
CðBCÞ#
CðBCÞ# AðBCÞ# B
! ð2:12Þ
;
where ðBCÞp ¼ I BCðBCÞ# . Proof. (i) Recall that, M # exists if and only if rðMÞ ¼ rðM2 Þ. Since AC ð1Þ C ¼ A, we can show that
rðMÞ ¼ r
A B C
2
rðM Þ ¼ r
! ¼r
0
0
B
C
0
A2 þ BC
AB
CA
CB
! ¼ rðCÞ þ rðBÞ;
!
¼r
BC
0
CA
CB
:
Therefore, it is obvious that the condition is sufficient. Now we show that condition is also necessary.
rðMÞ ¼ rðM 2 Þ ) rðCÞ þ rðBÞ ¼ r
BC
0
CA CB
6 rðBCÞ þ rð CA CB Þ 6 rðBCÞ þ rðCÞ 6 rðCÞ þ rðBÞ;
hence, rðBCÞ þ rðCÞ ¼ rðBÞ þ rðCÞ, which implies rðBÞ ¼ rðBCÞ. On the other hand,
rðCÞ þ rðBÞ ¼ r
BC CA
0 CB
6r
BC CA
þ rðCBÞ ¼ r
BC ð1Þ
CAC C
þ rðCBÞ 6 rðCÞ þ rðCBÞ 6 rðCÞ þ rðBÞ;
namely, rðCBÞ þ rðCÞ ¼ rðBÞ þ rðCÞ, means rðBÞ ¼ rðCBÞ. Therefore, rðBÞ ¼ rðCBÞ ¼ rðBCÞ, and rðBÞ 6 rðCÞ. Combining rðCÞ 6 rðBÞ shows that
rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ: (ii) From rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ, we know that ðBCÞ# and ðCBÞ# exist. Moreover, the following equalities hold
AðBCÞp ¼ 0;
CðBCÞp ¼ 0 and ðBCÞp B ¼ 0:
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Let the right hand side of (2.12) be N. A simple computation shows that
MN ¼
BCðBCÞ#
ðBCÞp AðBCÞ# B
0
CðBCÞ# B
! ¼ NM:
Furthermore, we can deduce that
MNM ¼ M
and NMN ¼ N:
According to the definition of group inverse, we know that N ¼ M # .
h
Remark that the condition AC ð1Þ C ¼ A in Theorem 2.4 can be replaced by RðA Þ # RðC Þ. And rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ is equivalent to rðBÞ ¼ rðCÞ ¼ rðCBCÞ ¼ rðBCBÞ. The following result is a special case of Theorem 2.4, which can be found in [4]. A B Corollary 2.7. Let M ¼ , where A; B 2 C nn , such that rðAÞ 6 rðBÞ. Then A 0 (i) M # exists if and only if rðBÞ ¼ rðAÞ ¼ rðBAÞ ¼ rðABÞ. (ii) If M# exists, then #
M ¼
ðBAÞp AðBAÞ#
ðBAÞ# B ðBAÞp AðBAÞ# AðBAÞ# B
AðBAÞ#
AðBAÞ# AðBAÞ# B
! ;
where ðBAÞp ¼ I BAðBAÞ# . Similarly, we can derive the following theorem, which can be regarded as a same case with the Theorem 1.2 in [25] given by Cao and Li, where they presented the necessary and sufficient conditions to the existence and the representation of the group AX A inverse for block matrix . B 0
Theorem 2.5. Let M be partitioned as (1.2), such that rðBÞ 6 rðCÞ. If BBð1Þ A ¼ A, or equivalently RðAÞ # RðBÞ, then
(i) M# exists if and only if rðBÞ ¼ rðCÞ ¼ rðBCÞ ¼ rðCBÞ. (ii) If M# exists, then
M ¼
!
ðBCÞ# AðBCÞp
ðBCÞ# B
CðBCÞ# CðBCÞ# AðBCÞ# AðBCÞp
CðBCÞ# AðBCÞ# B
#
;
where ðBCÞp ¼ I BCðBCÞ# . Corollary 2.8. Let M ¼
A C
A , where A; C 2 C nn , such that rðAÞ 6 rðCÞ. Then 0
(i) M# exists if and only if rðCÞ ¼ rðAÞ ¼ rðCAÞ ¼ rðACÞ. (ii) If M # exists, then #
M ¼
!
ðACÞ# AðACÞp
ðACÞ# A
CðACÞ# CðACÞ# AðACÞ# AðACÞp
CðACÞ# AðACÞ# A
;
where ðACÞp ¼ I ACðACÞ# . Based on Theorem 2.4 and Theorem 2.5, we can deduce the following results. Theorem 2.6. Let M be partitioned as (1.2). If AC ð1Þ C ¼ A, or equivalently RðA Þ # RðC Þ, and rðCÞ ¼ rðCBCÞ, then IndðMÞ 6 2, and
MD ¼
ðBCÞp AðBCÞ#
ðBCÞ# B ðBCÞp AðBCÞ# AðBCÞ# B
CðBCÞ#
CðBCÞ# AðBCÞ# B
where ðBCÞp ¼ I BCðBCÞ# .
! :
ð2:13Þ
X. Liu, H. Yang / Applied Mathematics and Computation 218 (2012) 8978–8986
8985
Proof. Consider the splitting
M¼P¼
A
C ð1Þ CB
C
0
!
0 B C ð1Þ CB
þ
0
! D
¼ P þ Q:
0
We observe that Q 2 ¼ 0 and PQ ¼ 0. Applying Lemma 1.4, we obtain
MD ¼ PD þ Q ðPD Þ2 :
ð2:14Þ
D
Now, we calculate P . Since rðCÞ ¼ rðCBCÞ, according to Lemma 1.2, it follows that rðCÞ ¼ rðBCÞ ¼ rðCBÞ ¼ rðCBCÞ, so P satisfies the conditions in Theorem 2.4 (i), that is to say P# exists. Moreover, it is clearly that C ð1Þ CBCðBCÞ# ¼ C ð1Þ C. Hence, by Theorem 2.4, we have
! C ð1Þ CðBCÞ# B F C AðBCÞ# AðBCÞ# B ; CðBCÞ# AðBCÞ# B
F C AðBCÞ# CðBCÞ#
#
P ¼
ð2:15Þ
where F C ¼ I C ð1Þ C. Substituting (2.15) into (2.14}) gives (2.13). It is easy to verify that
MM p ¼ MðI MMD Þ ¼
0 ðBCÞp B 0
!
0
;
which means that M 2 M p ¼ 0, i.e., IndðMÞ 6 2. h Theorem 2.7. Let M be partitioned as (1.2). If BBð1Þ A ¼ A, or equivalently RðAÞ # RðBÞ, and rðBÞ ¼ rðBCBÞ, then IndðMÞ 6 2, and
M ¼
!
ðBCÞ# AðBCÞp
ðBCÞ# B
CðBCÞ# CðBCÞ# AðBCÞ# AðBCÞp
CðBCÞ# AðBCÞ# B
D
;
where ðBCÞp ¼ I BCðBCÞ# . Remark. Although the expressions of M # in Theorem 2.4 (or Theorem 2.5) and M D in Theorem 2.6 (or Theorem 2.7) have the same form, we observe that their needful conditions are different, because the condition rðBÞ ¼ rðCÞ in Theorem 2.6 (or Theorem 2.7) is not satisfied.
3. Examples In this section, we give two examples to illustrate Theorem 2.3 and Theorem 2.7. A B Example 3.1. Consider the block matrix M ¼ , where C 0
0
1 0 1
B C A ¼ @ 0 1 0 A; 0
0
0
1
1
1
B B¼@ 1
0
1
C 1 A; 1
1
C¼
1 1 1
2 0 2
:
It is easy to verify that A2 ¼ A; CAp B ¼ 0 and ðCBÞ# exists. Furthermore, we can calculate that
0
1 0 0 1 B C A ¼ @ 0 0 0 A; 0 0 1 p
ðCBÞ# ¼
1 1 ; 0 0
ðCBÞp ¼
0 1 : 0 1
Hence, applying Theorem 2.3, we get
1 6 1 6 2 2 C B 1 C B 3 0 3 1 C B D B M ¼B 7 1 7 2 2 C C: C B 1 1 1 1 A @ 1 0
2
0
2
0
0
Example 3.2. Consider the block matrix M ¼
A C
B , where 0
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0
1 0 1
B C A ¼ @ 0 2 1 A; 0
0
0
1
1 1
1
B C B ¼ @ 0 1 A;
C¼
0 0
0
1
1 0
1 1 1
:
We observe that RðAÞ # RðBÞ, and rðBÞ ¼ rðBCBÞ. Furthermore, we can compute that
0 ðBCÞ# ¼
2 4 3
1B @2 4 0
0 0
1
C 1 A;
0 ðBCÞp ¼
0
0 0
1
1
1B C @ 0 0 1 A: 2 0 0 2
By Theorem 2.7, we have
1 0 0 3 2 2 C B 3 2 2 C B0 0 C 1B D 0 0 0 0 0 C M ¼ B C: 4B C B 4 4 1 2 6 A @ 0 4 4 4 4 0
Acknowledgements The authors would like to thank the editors and the reviewers for their valuable comments and helpful suggestions, which improved the paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11171361) and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20110191110033). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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