Filomat 26:2 (2012), 135–145
DOI (will be added later)
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
A comment on some recent results concerning the Drazin inverse of an
anti-triangular block matrix
Chun Yuan Denga
a School
of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China.
D
Abstract. In this
! note we give formulae for the Drazin inverse M of an anti-triangular special block matrix
A B
under some conditions expressed in terms of the individual blocks, which generalize some
C 0
recent results given by Changjiang Bu [7, 8] and Chongguang Cao [10], etc.
M=
1. Introduction
This research came up when we read some recent papers [7]-[10] which were concerned about calculating the Drazin inverses or group inverses of the anti-triangular special block matrices. The concept
of the Drazin inverse plays an important role in various fields like Markov chains, singular differential
and difference equations, iterative methods, etc. [1]-[6], [15]. Our purpose
is to give representations for
!
A B
the Drazin inverse of the anti-triangular block matrix M =
under some conditions expressed in
C 0
terms of the individual blocks. Block matrices of this form arise in numerous applications, ranging from
constrained optimization problems to the solution of differential equations [1], [2], [3], [13], [16], [17].
Let P = P2 be an idempotent
matrix.
C. Cao in !2006 [10] gave
of every! one of the seven
!
!
! the group ∗inverse
!
!
∗
∗
∗
PP P
P P
PP PP
P P
P PP
P PP∗
P∗ P
matrices:
,
,
,
,
,
and
.
P
0
PP∗ 0
P
0
P∗ 0
PP∗
0
P∗
0
P 0
Recently, C. Bu, et al. in [7–9] has
! obtained the new representations for the group inverse of a 2 × 2 antiA A
triangular matrix M =
, where A2 = A in terms of the group inverse of AB. In the present paper
B 0
we will find explicit expressions for the Drazin inverse of a 2 × 2 anti-triangular operator matrix M under
other weaker constraints. Our results generalize some recent results given by Changjiang Bu [7, 8] and
Chong Guang Cao [10], etc.
In this note, let A be an n × n complex matrix. We denote by N(A), R(A) and rank(A) the null space,
the range and the rank of matrix A, respectively. The Drazin inverse [2] of A ∈ Cn×n is the unique complex
2010 Mathematics Subject Classification. 15A09; 47A05.
Keywords. Anti-triangular matrix, Drazin inverse, Group inverse, Block matrix.
Received: 4 April 2011; Accepted: 20 August 2012
Communicated by Dragana S. Cvetković-Ilić
Partial support was provided by a grant from the Ph.D. Programs Foundation (No.20094407120001)
Email address: [email protected];[email protected] (Chun Yuan Deng)
136
C. Deng / Filomat 26:2 (2012), 135–145
matrix AD ∈ Cn×n satisfying the relations
AAD = AD A, AD AAD = AD , Ak AAD = Ak
for all k ≥ r,
(1)
where r = ind(A), called the index of A, is the smallest nonnegative integer such that rank(Ar+1 ) = rank(Ar ).
We will denote by Aπ = I − AAD the projection on N(Ar ) along R(Ar ). In the case ind(A) = 1, AD reduces to
the group inverse of A, denoted by A# . In particular, A is nonsingular if and only if ind(A) = 0.
2. Key lemmas
In this section, we state some lemmas which will be used to prove our main results.
Lemma 2.1. (see [7, Lemma 2.5]) Let A, B ∈ Cn×n such that rank(A) = r. If A2 = A and rank(B) = rank(BAB),
then A and B can be written as
!
!
I 0
B1
B1 X
and B =
A=
0 0
YB1 YB1 X
with respect to space decomposition Cn = R(A)⊕N(A), where AB, BA and B1 ∈ Cr×r are group invertible, X ∈ Cr×(n−r)
and Y ∈ C(n−r)×r .
The following lemma concerns the Drazin inverse of 2 × 2 block matrix.
Lemma 2.2. (see Lemma 2.2 and Corollary 2.3 in [14]) Let M =
A
C
B
D
!
such that D is nilpotent and
ind(D) = s. If BC = 0 and BD = 0, then

AD

 s−1
D
M =  P i

D C(AD )i+2
i=0



s−1
P i
.
D i+3
D C(A ) B 
(AD )2 B
i=0
Lemma 2.3. (see [11, Theorem 2.3]) Let A, B ∈ Cn×n such that AB = BA. Then
(1) (AB)D = BD AD = AD BD .
(2) ABD = BD A and AD B = BAD .
(3) (AB)π = Bπ when A is invertible.
!
A A
Lemma 2.4. Let M =
such that A is nilpotent and ind(A) = s. If BA = 0, then M is nilpotent with
B 0
ind(M) ≤ s + 1.
!
As+1 + As B As
s+1
Proof. Note that, if BA = 0, then M =
= 0.
0
0
!
A B
Let M =
, where A ∈ Cd×d , B ∈ Cd×(n−d) and C ∈ C(n−d)×d . N. Castro-González and E. Dopazo
C 0
(see [3, Theorem 4.1]) had proved that, if CAD A = C and AD BC = BCAD , then (see [3], pp.267)
i
h
i
 D2

 A ) [W1 + (AD )2 BCW2 (BC)π A

(BC)D + (AD )2 W1 (BC)π B
D
h
i
h
i  ,
M = 
(2)
D
D 2
π
D 2
D 3
π
C (BC) + (A ) W1 (BC)
C −A((BC) ) + (A ) W2 (BC) B 
137
C. Deng / Filomat 26:2 (2012), 135–145
where
h
i
r = ind (AD )2 BC ,
W1 =
r−1
X
(−1)j C(2j + 1, j)(AD )2j (BC)j ,
r−1
X
W2 =
j=0
(−1)j C(2j + 2, j)(AD )2j (BC)j .
j=0
As a directly application of [3, Theorem 4.1]) and Lemma 2.3, we get the following result.
!
A A
Lemma 2.5. Let M =
such that A is nonsingular and ind(B) = r. If BA = AB, then
B 0
!
π
W1 Bπ + W2 BB
BD + W1 Bπ
D
h
i
M =
,
BBD + W1 BBπ A−1
−BD + W2 BBπ
where
r−1
X
W1 =
(−1) j C(2j + 1, j)A− j−1 B j
r−1
X
and W2 =
j=0
(−1) j C(2j + 2, j)A− j−2 B j .
j=0
In Lemma 2.5, if A = I, then
I
B
where Y2 = W1 =
r−1
P
I
0
!D
Y1 Bπ
BBD + Y2 BBπ
=
(−1) j C(2j + 1, j)B j
BD + Y2 Bπ
−BD + (Y1 − Y2 )Bπ
and Y1 Bπ = Y2 Bπ + W2 BBπ =
j=0
!
,
r−1
P
(−1) j C(2j, j)B j Bπ . This result had
j=0
been given by N. Castro-González and E. Dopazoin in their celebrated paper [3, Theorem 3.3].
3. Main results
!
A A
Our first purpose is to obtain a representation for M of the matrix M =
under some conditions,
B 0
where A, B are n × n matrices. Throughout
our development, we will be concerned with the anti-upper!
A B
triangular matrix M =
. However, the results we obtain will have an analogue for anti-lowerC 0
!
0 A
triangular matrix M =
. The following result generalizes the recent result given by Changjiang
C B
Bu, et al (see [7, Theorem 3.1]).
!
A A
and e
B = (I − Aπ )B(I − Aπ ) with ind(A) = s and ind(e
B) = r. If
Theorem 3.1. Let M =
B 0
D
BAAπ = 0 and
then
M
D

s

P
= R +
i=0
where
AAπ
Aπ BAπ
AAπ
0


Γ1 e
Bπ + Γ2e
Bie
Bπ
R =  heeD
π
BB + Γ 1 e
Be
B AD
!i
(I − Aπ )(BA − AB)(I − Aπ ) = 0,
0
0
Aπ B(I − Aπ ) 0
!
 "

 × I + R
i+2 

R
0
(I − Aπ )BAπ
0
0
!#
,
(3)

e
BD + Γ1 e
Bπ 
,
−e
BD + Γ2 e
Be
Bπ 
(4)
Γ1 =
r−1
P
(−1) j C(2j + 1, j)(AD ) j+1 e
B j,
j=0
Γ2 =
r−1
P
j=0
(−1) j C(2j + 2, j)(AD ) j+2e
B j.
138
C. Deng / Filomat 26:2 (2012), 135–145
Proof. Let X1 = N(Aπ ) and X2 = R(Aπ ). Then X = X1 ⊕ X2 . Since A is ind(A) = s, A has the form
As2 = 0 and
A = A1 ⊕ A2 with A1 nonsingular,
AD = A−1
⊕ 0.
1
(5)
Using the decomposition X ⊕ X = X1 ⊕ X2 ⊕ X1 ⊕ X2 , we have

 A1
 0

M = 
 B1

B4
0
A2
B3
B2
A1
0
0
0
0
A2
0
0




 X1 
  X1 
 X 
  X2 
 2 

 
  X1  −→  X1  .



 
X2
X2
(6)
!
0 I
Define I0 = I ⊕
⊕ I. It is clear that I0 , as a matrix from X1 ⊕ X2 ⊕ X1 ⊕ X2 onto X1 ⊕ X1 ⊕ X2 ⊕ X2 , is
I 0
nonsingular with I0 = I0∗ = I0−1 . Hence
M
D


 I 0 0 0   A1
 0 0 I 0   B
  1

= 

 0 I 0 0   0


B4
0 0 0 I
A1
0
0
0
0
B3
A2
B2
0
0
A2
0
D

  I 0 0 0 
  0 0 I 0 

 
 = I0
 
  0 I 0 0 


0 0 0 I
A0
C0
B0
D0
!D
I0 ,
(7)
where
A0 =
A1
B1
A1
0
!
,
B0 =
0
B3
0
0
!
,
C0 =
0
B4
0
0
!
,
D0 =
A2
B2
A2
0
!
.
(8)
If (I − Aπ )(BA − AB)(I − Aπ ) = 0, using the representations in (5) and (6), we get A1 is nonsingular and
A1 B1 = B1 A1 . Since ind[(I − Aπ )B(I − Aπ )] = ind[B1 ] = r, by Lemma 2.5, we get
R =:
I0
A0
0
0
0
!D
I0
=

π
π
 h W1 B1 + W2 B1iB1

 B BD + W1 B1 Bπ1 A−1
1
I0  1 1

0

0
=

W1 Bπ1 + W2 B1 Bπ1


0
 h
i

 B1 BD + W1 B1 Bπ A−1
1
1

1
0
=

eπ
eeπ

 h Γ1 B + Γ2 BiB
 e
D
π
e
e
e
BB + Γ1 BB AD
BD
+ W1 Bπ1
1
−BD
+ W2 B1 Bπ1
1
0
0
0
0
0
0
BD
+ W1 Bπ1
1
0
D
−B1 + W2 B1 Bπ1
0
e
BD + Γ 1 e
Bπ
D
e
−B + Γ2e
Be
Bπ
0
0
0
0
0
0
0
0
0
0
0
0




 I0











 ,

where
W1 =
Γ1 =
r−1
P
j=0
r−1
P
− j−1
(−1) j C(2j + 1, j)A1
j
B1 ,
(−1) j C(2 j + 1, j)(AD ) j+1 e
B j,
j=0
W2 =
Γ2 =
r−1
P
j=0
r−1
P
j=0
− j−2
(−1) j C(2j + 2, j)A1
j
B1 ,
(−1) j C(2j + 2, j)(AD ) j+2e
B j.
139
C. Deng / Filomat 26:2 (2012), 135–145
Since BAAπ = 0, we get B3 A2 = 0 and B2 A2 = 0. By Lemma 2.4, we get D0 is nilpotent with ind(D0 ) ≤ s + 1.
Note that B3 A2 = 0 implies that B0 C0 = 0 and B0 D0 = 0. By Lemma 2.2, we obtain


!D
AD
(AD
)2 B 0


0
0
A
B
 s
0
0
s
 I
P
I0 = I0  P
MD = I0
i
i
D i+2
D i+3
D0 C0 (A0 )
D0 C0 (A0 ) B0  0
C0 D0

i=0
"
= I0
AD
0
0

s

P
= R +
0
0
!
+
s
P
0
0
i=0
AAπ
Aπ BAπ
i=0
AAπ
0
0
Di0
!i
i=0
!
0
C0
!
0
0
(AD
)i+2
0
0
!
0
0
Aπ B(I − Aπ ) 0
0
0
!#
"
× I+
 "

 × I + R
i+2 

R
AD
0
0
0
0
!
0
(I − Aπ )BAπ
0
0
B0
0
0
0
!#
I0
(9)
!#
.
We remark that, from the above theorem we get the following corollaries.
!
A A
.
Corollary 3.2. Let M =
B 0
(i) If AB = BA, ind(A) = 1 and ind(B) = r, then
D
M =
where
Γ1 =
r−1
P
π
π
h Γ1 B + Γ2 BB
i
D
π
BB + Γ1 BB AD
(I − Aπ )BD + Γ1 Bπ
−(I − Aπ )BD + Γ2 BBπ
(−1) j C(2 j + 1, j)(A# ) j+1 B j ,
Γ2 =
j=0
r−1
P
!
,
(−1) j C(2j + 2, j)(A# ) j+2 B j .
j=0
(ii) If A, B are group invertible and AB = BA, then
D
M =
A# Bπ
[I − Bπ ] A#
(I − Aπ )B# + A# Bπ
−(I − Aπ )B#
!
.
In addition, if Aπ B = 0, then Aπ B# = 0, MD becomes the group inverse and
!
A# Bπ
B# + A# Bπ
#
M =
.
[I − Bπ ] A#
−B#
(iii) If A, B are invertible, then
M−1 =
A
B
Corollary 3.3. Let M =
R=
X=
h
r−1
P
j=0
!
B−1
−B−1
.
!
A
, where A, B ∈ Cn×n , A = A2 and ind(ABA) = r. Then
0
(i) (see [9, Theorem 3.2])
"
! #"
0
0
D
M = R+
R2 I + R
(I − A)BA 0
where
0
A−1
X+Y
i
(AB)D + X ABA
!#
0
0
AB(I − A) 0
(AB)D A + X
−(AB)D A + Y
(−1) j C(2j + 1, j)(AB)π (AB) j A,
,
(10)
!
,
Y=
r−1
P
j=0
(−1) j C(2j + 2, j)(AB)π (AB) j+1 A.
140
C. Deng / Filomat 26:2 (2012), 135–145
(ii) (see [7, Theorem 3.1]) M# exists if and only if rank(B) = rank(BAB) and
A − (AB)# + (AB)# A − (AB)# ABA
(BA)# B − (BA)# (AB)# AB − (BA)#
M# =
A + (AB)# A + (AB)# ABA
−(BA)#
!
.
(11)
Proof. (i) If A = A2 , we have ind(A) = 1, A = AD , Aπ = I − A,
e
BD = [(I − Aπ )B(I − Aπ )]D = (ABA)D = AB[(AAB)D ]2 A = (AB)D A
and
So
e
B j = (ABA) j = (AB) j A = A(BA) j .
e
Bπ = (ABA)π = I − (ABA)D (ABA) = I − (AB)D ABA = I − A + (AB)π A = I − A + A(BA)π .
Hence, Γ1 and Γ2 in (3.2) reduce as
Γ1 =
Γ2 =
r−1
P
r−1
P
j=0
j=0
(−1) j C(2j + 1, j)(AD ) j+1e
Bj =
r−1
P
r−1
P
j=0
j=0
Bj =
(−1) j C(2j + 2, j)(AD ) j+2e
(−1) j C(2j + 1, j)(AB) j A,
(−1) j C(2j + 2, j)(AB) j A.
Let
r−1
P
X = Γ1 e
Bπ = (−1) j C(2j + 1, j)(AB)π (AB) j A,
r−1
P
Y = Γ2 e
Be
Bπ = (−1) j C(2j + 2, j)(AB)π (AB) j+1 A.
j=0
Then R in (3.2) reduces as


Γ1 e
Bπ + Γ2e
Bie
Bπ
R =  heeD
BB + Γ1e
Be
Bπ AD
j=0
e
BD + Γ 1 e
Bπ
−e
BD + Γ 2 e
Be
Bπ


 =

h
X+Y
i
(AB)D + X ABA
(AB)D A + X
−(AB)D A + Y
!
.
By Theorem 3.1, we get
D
M

1

P
= R +
i=0
"
=
R+
0
0
(I − A)B(I − A) 0
0
0
(I − A)BA 0
!
!i
#"
R
2
I+R
0
0
(I − A)BA 0
!
0
0
AB(I − A) 0
 "

 × I + R
i+2 

R
0
0
AB(I − A) 0
!#
!#
.
(ii) See Theorem 3.1 in [7] for the proof that M# exists if and only if rank(B) = rank(BAB). By Lemma 2.1,
we have ind(ABA) ≤ 1, AB and BA are group invertible. So, by item (i), we get X = (AB)π A, Y = 0,
!
(AB)π A
(AB)# A + (AB)π A
R=
(AB)# ABA (AB)# A − (AB)# A
and
2
R =
(AB)π A + (AB)# A (AB)π A − [(AB)# ]2 A
−(AB)# A
(AB)# A + [(AB)# ]2 A
!
.
Thus, collecting the above computations in the expression (10) for MD , we get the statement of (11). 141
C. Deng / Filomat 26:2 (2012), 135–145
Note that
r−1
X
(−1) j C(2j, j)B j Bπ =
j=0
r−1
X
(−1) j C(2j + 1, j)B j Bπ +
j=0
In Corollary 3.3, if we set A = I and Z =
r−1
X
(−1) j C(2j + 2, j)B j+1 Bπ .
j=0
r−1
P
(−1) j C(2j, j)B j Bπ , then we get Y = Z − X and Corollary 3.3 (resp.
j=0
Theorem 3.1) reduces as the following result which had been given in [3].
!
I I
, where B ∈ Cn×n and ind(B) = r. Then
Corollary 3.4. ([3, Theorem 3.3]) Let M =
B 0
!
Z
BD + X
D
M =
,
BD B + XB
−BD + Z − X
where X =
r−1
P
(−1) j C(2j + 1, j)B j Bπ ,
Z=
j=0
r−1
P
(−1) j C(2 j, j)B j Bπ .
j=0
Our next !purpose is to obtain a representation for the Drazin inverse of block antitriangular matrix
A B
M=
, where A, C ∈ Cn×n , which in some different ways generalizes recent results given in [10, 14].
C 0
We start introducing a different method to give matrix block representation. Let S = −CAD B, ind(A) = m
and ind(S) = n. In (5) and (6), if we set
X1 = N(Aπ ),
X2 = R(Aπ ),
Y1 = N(Sπ ) and
Y2 = R(Sπ ).
Then X ⊕ Y = X1 ⊕ X2 ⊕ Y1 ⊕ Y2 . In this case, A and S have the forms
A = A1 ⊕ A2 with A1 nonsingular,
S = S1 ⊕ S2 with S1 nonsingular,
Am
= 0 and
2
Sn2 = 0
AD = A−1
⊕ 0,
1
(12)
⊕ 0.
and SD = S−1
1
Using the decomposition X ⊕ Y = X1 ⊕ X2 ⊕ Y1 ⊕ Y2 , we have





 A1 0 B1 B3   X1 
 X1 
 0 A B B   X 



2
4
2 
  2  −→  X2  .
M = 




0   Y1 
 C1 C3 0
 Y1 





C4 C2 0
0
Y2
Y2
Note that the generalized Schur complement
!
C1 C3
D
S = S1 ⊕ S2 = −CA B = −
C4 C2
A−1
1
0
0
0
!
B1
B4
(13)
B3
B2
!
=
−C1 A−1
B1
1
−C4 A−1
B
1
1
−C1 A−1
B3
1
−C4 A−1
B
3
1
!
.
Comparing the two sides of the above equation, we have
−1
S1 = −C1 A−1
S2 = −C4 A−1
C1 A−1
1 B1 ,
1 B3 ,
1 B3 = 0 and C4 A1 B1 = 0.
!
0 I
In this case, I0 = I ⊕
⊕ I as a matrix from X1 ⊕ X2 ⊕ Y1 ⊕ Y2 onto X1 ⊕ Y1 ⊕ X2 ⊕ Y2 is nonsingular
I 0
with I0 = I0∗ = I0−1 . Hence
M
D

 A1
 C

= I0  1
 0

C4
B1
0
B4
0
0
C3
A2
C2
B3
0
B2
0
D


 I0 := I0


A0
C0
B0
D0
!D
I0 ,
(14)
142
C. Deng / Filomat 26:2 (2012), 135–145
where
A0 =
A1
C1
!
B1
0
,
0
C3
B0 =
!
B3
0
,
0
C4
C0 =
B4
0
!
,
A2
C2
D0 =
B2
0
!
.
(15)
Since the Schur complement of A1 in A0 is −C1 A−1
B1 = S1 and S1 is nonsingular, it follows that A0 is
1
nonsingular with
A−1
0 =
Let R = I0
R=
A−1
+ A−1
B1 S−1
C1 A−1
1
1
1
1
−1
−1
−S1 C1 A1
A−1
0
0
0
0
!
−A−1
B1 S−1
1
1
−1
S1
.
(16)
!
I0 . Using the rearrangement effect of I0 , we get
AD + AD BSD CAD
−SD CAD
!
−AD BSD
SD
.
(17)
The expression (17) is called the generalized-Banachiewicz-Schur form of the matrix M and can be found
in some recent papers [14].
Now, we are in position to prove the following theorem which provides expressions for MD .
Theorem 3.5. Let M =
(I − Sπ )CAπ B = 0,
A B
C 0
!
and S = −CAD B with ind(A) = m. If
(I − Sπ )CAπ A = 0,
(I − Aπ )BSπ C = 0,
BSπ CAπ = 0,
(18)
then

m+1

P
M = R +
D
i=0
AAπ
Sπ CAπ
!i
Aπ BSπ
0
0
Sπ C(I − Aπ )
Aπ B(I − Sπ )
0
!
 "

 × I + R
i+2 

R
0
(I − Sπ )CAπ
(I − Aπ )BSπ
0
where R is defined as in (17).
Proof. Let A0 , B0 , C0 and D0 be defined by (15). Similar to the proof of Theorem 3.1, it is trivial to check
that the conditions in (18) imply that B0 C0 = 0 and B0 D0 = 0. Note that
A2
0
π
B2
0
!k
=
Ak2
0
Ak−1
B2
2
0
π
The condition BS CA = 0 implies that B2 C2 = 0 and
Dm+2
0
=
A2
C2
B2
0
=
A2
0
B2
0
!m+2
"
=
!m+2
+
A2
0
0
C2
B2
0
0
0
!
!
+
0
C2
A2
0
B2
0
!
= 0 for
A2
0
0
0
B2
0
!
k ≥ m + 1.
0
C2
0
0
!
= 0. So
!#m+2
(19)
!m+1
= 0.
!#
.
143
C. Deng / Filomat 26:2 (2012), 135–145
D0 is nilpotent and ind(D0 ) ≤ m + 2. By Lemma 2.2 and the proof in (9), we obtain


!D
AD
(AD
)2 B 0


0
0


A
B
0
0
m+1
P i
P i
I
MD = I0
I0 = I0  m+1
D
i+2
D
i+3
C0 D0
D0 C0 (A0 )
D0 C0 (A0 ) B0  0

i=0
i=0
"
= I0
AD
0
0

m+1

P
= R +
i=0
"
× I+R
0
0
!
+
m+1
P
i=0
AAπ
Sπ CAπ
0
0
0
Di0
Aπ BSπ
0
0
(I − Sπ )CAπ
!
!i
0
C0
0
0
!
(AD
)i+2
0
0
!#
"
× I+
Aπ B(I − Sπ )
0
0
Sπ C(I − Aπ )
(I − Aπ )BSπ
0
0
0
!
AD
0
0
0
0
!
0
0
B0
0
!#
I0



i+2 

R
!#
.
We remark that our result has generalized
! some results in the literature. In [10], Chong Guang Cao has
PP∗ P
given the group inverse of M =
, where P is an idempotent. Note that
P
0
(PP∗ )D = (PP∗ )# = (PP∗ )+
and PP∗ (PP∗ )D P = PP∗ (PP∗ )+ P = P.
If A = PP∗ , B = C = P in Theorem 3.5, then
S = −CAD B = −P(PP∗ )D P = −(PP∗ )D P,
It follows that
Aπ B = 0,
Aπ A = 0,
BSπ = 0,
and R in (17) reduces as
R=
AD + AD BSD CAD
−SD CAD
SD = −PP∗ P,
−AD BSD
SD
!
=
Sπ = I − P.
Sπ C = 0
0
PP∗ (PP∗ )D
P
−PP∗ P
!
.
By a direct computation we get the following result.
Corollary 3.6. [10, Theorem 2.1] Let P be an idempotent matrix and M =
M#
"
= R I+R
0
P[I − PP∗ (PP∗ )D ]
0
0
!#
=
PP∗
P
!
P
. Then
0
PP∗ (I − P)
(PP∗ )2 (P − I) + P
P
−PP∗ P
!
.
In Corollay 3.6, the reason that MD is replaced by M# is M satisfies the relation MMD M = M. Similar to
Corollary 3.6, if M is the matrix from the set
(
!
!
!
!
!)
P P
PP∗ PP∗
P P
P PP∗
P PP∗
,
,
,
,
,
PP∗ 0
P
0
P∗ 0
PP∗
0
P∗
0
then M satisfies Theorem 3.5. Hence, Theorem 2.1–Theorem 2.6 in [10] are all the special cases of our
Theorem 3.5.
If A in Theorem 3.5 is nonsingular and A−1 BC is group invertible, then Aπ = 0 and ind(A) = 0 and
0
=
BC(A−1 BC)π = BC − BC(A−1 BC)D A−1 BC = BC − BCA−1 BC[(A−1 BC)D ]2 A−1 BC
=
h
i
h
i
B I − CA−1 BC[(A−1 BC)D ]2 A−1 B C = B I − CA−1 B(CA−1 B)D C = BSπ C.
From the above computations, we get the conditions in (18) hold and Theorem 3.5 reduces as the following:
144
C. Deng / Filomat 26:2 (2012), 135–145
Corollary 3.7. Let M =
!
A B
, A be nonsingular, S = −CA−1 B such that A−1 BC is group invertible, then
C 0
"
! # "
!#
0
0
0 BSπ
2
MD = R +
R
×
I
+
R
.
Sπ C 0
0
0
Finally, we derive from Theorem 3.5 some particular representations of AD under certain additional
conditions.
!
A B
and S = −CAD B with ind(A) = m. Let R be defined as in (17).
Corollary 3.8. Let M =
C 0
(i) If C(I − AAD ) = 0 and the generalized Schur complement S = −CAD B is nonsingular, then
MD = R +
m+1
P
i=0
AAπ
0
0
0
!i
0
0
Aπ B
0
!
Ri+2 .
(ii) (see [14, Theorem 1.1]) If C(I − AAD ) = 0, (I − AAD )B = 0 and the generalized Schur complement S = −CAD B
is nonsingular, then
!
AD + AD BS−1 CAD −AD BS−1
.
MD =
−S−1 CAD
S−1
(iii) (see [14, Theorem 3.1 or Corollary 3.2 ]) If C(I − AAD )B = 0, C(I − AAD )A = 0 and the generalized Schur
complement S = −CAD B is nonsingular, then
"
!
# "
!#
m−1
P 0 Ai Aπ B
0
0
D
i+1
M
= I+
R
R I+R
.
CAπ 0
0
0
i=0
In Corollary 3.8(iii), if CAπ = 0, Aπ B = 0 and the generalized Schur complement S = −CAD B is
nonsingular,, then MD = R, which is famous Banachiewicz-Schur formula.
4. Acknowledgments
The authors thank Prof. Dragana S. Cvetković-Ilić and the referee for their very useful and detailed
comments which greatly improve the presentation.
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