Several expressions for the generalized Drazin inverse of a block matrix in a Banach algebra

Applied Mathematics and Computation 220 (2013) 374–381

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Several expressions for the generalized Drazin inverse of a block matrix in a Banach algebra Dijana Mosic´ ⇑, Dragan S. Djordjevic´ 1 Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18000 Niš, Serbia

a r t i c l e

i n f o

Keywords: Generalized Drazin inverse Schur complement Block matrix

a b s t r a c t We present some new representations for the generalized Drazin inverse of a block matrix with generalized Schur complement being generalized Drazin invertible in a Banach algebra under conditions weaker than those used in recent papers on the subject. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Let A be a complex unital Banach algebra with unit 1. For a 2 A, the symbols rðaÞ and qðaÞ will denote the spectrum and the resolvent set of a, respectively. We use Anil and Aqnil , respectively, to denote the sets of all nilpotent and quasinilpotent elements (rðaÞ ¼ f0g) of A. The concept of the generalized Drazin inverse in Banach algebras was introduced by Koliha (see [16]). In [13], Harte presented an alternative definition of a generalized Drazin inverse in a ring. For a 2 A, if there exists an element b 2 A which satisfies

bab ¼ b;

ab ¼ ba;

a  a2 b 2 Aqnil ;

then b is called the generalized Drazin inverse of a (or Koliha–Drazin inverse of a), and a is generalized Drazin invertible. If the generalized Drazin inverse of a exists, it is unique and denoted by ad . The set of all generalized Drazin invertible elements of A is denoted by Ad . If a 2 Ad , the spectral idempotent ap of a corresponding to the set f0g is given by ap ¼ 1  aad . The Drazin inverse is a special case of the generalized Drazin inverse for which a  a2 b 2 Anil . Obviously, if a is Drazin invertible, then it is generalized Drazin invertible. The group inverse is the Drazin inverse for which the condition a  a2 b 2 Anil is replaced with a ¼ aba. We use a# to denote the group inverse of a, and we use A# to denote the set of all group invertible elements of A. We need the following important result from [3]. Lemma 1.1 [3, Lemma 2.4]. Let b; q 2 Aqnil and let qb ¼ 0. Then q þ b 2 Aqnil . The next result is proved for matrices [15, Theorem 2.1], for bounded linear operators [12, Theorem 2.3] and for elements of Banach algebra [3]. Lemma 1.2 [3, Example 4.5]. Let a; b 2 Ad and let ab ¼ 0. Then

⇑ Corresponding author. 1

E-mail addresses: [email protected] (D. Mosic´), [email protected] (D.S. Djordjevic´). The authors are supported by the Ministry of Education and Science, Republic of Serbia, Grant No. 174007.

0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.06.018

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 220 (2013) 374–381

d

ða þ bÞ ¼

375

1 1 X X nþ1 d nþ1 p n ðb Þ an ap þ b b ðad Þ : n¼0

n¼0

If a 2 Aqnil , then ad exists and ad ¼ 0. Consequently, by Lemma 1.2, the following lemma, which the part (i) is proved by Castro–González and Koliha [3] and part (ii) for bounded linear operators in [12, Theorem 3.2], holds. Lemma 1.3. Let b 2 Ad and a 2 Aqnil . d

(i) [3, Corollary 3.4] If ab ¼ 0, then a þ b 2 Ad and ða þ bÞ ¼ P d d nþ1 n (ii) If ba ¼ 0, then a þ b 2 Ad and ða þ bÞ ¼ 1 . n¼0 a ðb Þ

P1

d nþ1 n a . n¼0 ðb Þ

Recall that, if p ¼ p2 2 A is an idempotent, we can represent element a 2 A as

 a¼

a11

a12

a21

a22

 ;

where a11 ¼ pap, a12 ¼ pað1  pÞ; a21 ¼ ð1  pÞap; a22 ¼ ð1  pÞað1  pÞ. Let

x¼



a b c

 2A

d

ð1Þ

relative to the idempotent p 2 A; a 2 ðpApÞd and let the generalized Schur complement xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd . The Drazin inverse is very useful, and has various applications in singular differential or difference equations, Markov chains, cryptography, iterative method and numerical analysis (see [6,7]). The problem of finding an explicit representation for the Drazin inverse of a complex block matrix in terms of its blocks was first proposed by Campbell and Meyer [5]. This problem is quite complicated, and there was no explicit formula for the Drazin inverse of a block matrix. Some special cases have been considered in [4,9,14,18,22–24]. The generalized Schur complement plays an important role in the representations for the Drazin inverse of a block matrix. Miao [18] and Wei [23] have been studied the Drazin inverse of a block matrix with the generalized Schur complement being nonsingular or it is equal to zero. The following result is well-known for complex matrices (see [18]) and it is proved for elements of Banach algebra [19].   a b d Lemma 1.4 [19, Lemma 2.2]. Let x ¼ 2 A relative to the idempotent p 2 A; a 2 ðpApÞd and let w ¼ aad þ ad bca be such c d that aw 2 ðpApÞd . If cap ¼ 0; ap b ¼ 0 and the generalized Schur complement xs ¼ d  cad b is equal to 0, then x 2 Ad and xd ¼ m, where

m¼



p

0

cad

0

"

2

½ðawÞd  a 0 0

#"

0

p

ad b

0

0

#

2

3 d 2 d 2 ½ðawÞ  a ½ðawÞ  b 5: ¼4 2 2 cad ½ðawÞd  a cad ½ðawÞd  b

ð2Þ

We use the next lemma which is proved in [20]. The expression (3) is called the generalized Banachiewicz–Schur form of xd . For more details see [1,2,4,8,14,21]. Lemma 1.5 [20, Lemma 2.1]. Let x be defined as in (1). Then the following statements are equivalent: (i) x 2 Ad and xd ¼ r, where

" r¼

ad þ ad bðxs Þd cad

ad bðxs Þd

ðxs Þd cad

ðxs Þd

(ii) ap b ¼ bðxs Þp ;

ðxs Þp c ¼ cap

and

# ð3Þ

; y¼



aap cap

 bðxs Þp 2 Aqnil . xs ðxs Þp

Hartwig et al. [14] presented formulae for the Drazin inverse of a 2  2 block matrix which involve a matrix in the form (3) when the generalized Schur complement is nonsingular, and a matrix in the form (2) when it is equal to zero, under the assumptions CAp B ¼ 0 and AAp B ¼ 0. Under different conditions and the hypothesis the Schur complement is either nonsingular or zero, these results are generalized in [17,24]. Deng et al. [11] obtained the explicit representations for the generalized Drazin inverse of operator matrix under the cases that the generalized Schur complement is either nonsingular or zero and (i) Ap BC ¼ 0; CAp B ¼ 0; Ap AB ¼ Ap BD, or (ii) BCAp ¼ 0; CAp B ¼ 0; CAp A ¼ DCAp .

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 220 (2013) 374–381

376

Castro-González and Martínez-Serrano [4] investigated conditions under which the Drazin inverse of a block matrix having generalized Schur complement group invertible, can be expressed in terms of a matrix in the Banachiewicz–Schur form and its powers. In [9], Deng and Wei introduced several explicit representations for the Drazin inverse of block-operator matrix with Drazin invertible Schur complement under certain circumstances. We introduce new explicit expressions for the generalized Drazin inverse of a block matrix x defined in (1) when the generalized Schur complement is generalized Drazin invertible. Expressions presented in Section 2 contains a matrix in the form p s (2) under the conditions ap bc ¼ 0 and xs c ¼ 0 (or bca ¼ 0 and bx ¼ 0). In Section 3 we derive formulae for the generalized d Drazin inverse x which involve the generalized Banachiewicz–Schur form (3). Thus, we extend some results in [11,19,23] to more general settings. 2. Expressions for the generalized Drazin inverse In the following theorem we derive a formula for the generalized Drazin inverse of block matrix x in (1) under some conditions. This formula is rather cumbersome and complicated but the theorem itself will have a number of useful consequences. Theorem 2.1. Let x be defined as in (1) where a 2 ðpApÞd and the generalized Schur d xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd ; m be defined as in (2) and let w ¼ aad þ ad bca be such that aw 2 ðpApÞd . If

ap bc ¼ 0 and xs c ¼ 0;

complement

ð4Þ

d

then x 2 A and

" #!   X 1 1 X 0 0 p 0 aap nþ1  x ¼ m d nþ2 s p n p s 0 ðx Þ cap ca a b½ðx Þ  n¼0 n¼0 0 " # 2 0 ap b½ðxs Þd   ðawÞd bðxs Þd þ 0 ðxs Þd  cad ðawÞd bðxs Þd 2 3 nþ2 1 X 0 ða  ðawÞd bcÞan1 ap b½ðxs Þd  4 5 þ nþ2 n¼1 0 ðð1  pÞ  cad ðawÞd bÞcan1 ap b½ðxs Þd  02 3 nþ1 1 X 0 aad ðawÞn1 ðawÞp b½ðxs Þd  5 @ 4 þ nþ1 n¼1 0 cad ðawÞn1 ðawÞp b½ðxs Þd  2 31 kþnþ2 k1 p n1 p 1 d s d X 0 aa ðawÞ ðawÞ bca a b½ðx Þ  5A: þ 4 kþnþ2 k1 k¼1 0 cad ðawÞn1 ðawÞp bca ap b½ðxs Þd  d

ap b x

n

s

ð5Þ

Proof. Suppose that x ¼ y þ z, where

"

y¼

a2 ad caad

aad b cad b

#

and z ¼



aap

ap b

cap

xs

 ð6Þ

:

Then zy ¼ 0, by ap ad ¼ 0 and (4). # In order to show that y 2 Ad , let Ay  a2 ad , By  aad b; C y  caad and Dy  cad b. Since ða2 ad Þ ¼ ad , then # # # Ay 2 ðpApÞ# ; Apy By ¼ ap aad b ¼ 0; C y Apy ¼ caad ap ¼ 0; ys ¼ Dy  C y A# B ¼ 0 and W ¼ A A þ A B C A ¼ w. Using Lemma y y y y y y y y y 1.4, we deduce that y 2 Ad and yd ¼ m. If      

z1 ¼

0

0

0 x

s

;

z2 ¼

0 ap b

0

and z3 ¼

0

aap

0

cap

0

;

we have z ¼ z1 þ z2 þ z3 ; z2 z3 ¼ 0; z3 z1 ¼ 0; z22 ¼ 0 and z1 ðz2 þ z3 Þ ¼ 0. From aap 2 ðpApÞqnil and rðz3 Þ # rpAp ðaap Þ [ rð1pÞAð1pÞ ð0Þ, we conclude that z3 2 Aqnil . By z2 2 Anil and Lemma 1.1, z2 þ z3 2 Aqnil . Observe that z1 2 Ad and Lemma 1.3 (ii) imply z 2 Ad and

zd ¼ zd1 þ

1 1 1 X X X nþ1 nþ2 d nþ1 ðz2 þ z3 Þn ðzd1 Þ ¼ zd1 þ zn1 ¼ zd1 þ zn3 z2 ðzd1 Þ : 3 ðz2 þ z3 Þðz1 Þ n¼1

n¼1 d

n¼0

d

Now, by Lemma 1.2, x 2 A and x ¼ x1 þ x2 , where

x1 ¼

1 X nþ1 ðyd Þ zn zp n¼0

and x2 ¼

1 X n¼0

yp yn ðzd Þ

nþ1

:

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 220 (2013) 374–381

The equality d

zz ¼ ðz1 þ

z2 Þzd1

1 X d nþ2 þ znþ1 ¼ 3 z2 ðz1 Þ n¼0

gives

x1 ¼

"

1 X nþ1 ðyd Þ n¼0

"

0 ap bðxs Þd 0

þ

xs ðxs Þd

" nþ2 # 1 X 0 anþ1 ap b½ðxs Þd  n¼0

0

can ap b½ðxs Þd 

nþ2

# "   X ! nþ2 #! 1 1 1  X X 0 0 p 0 0 anþ1 ap b½ðxs Þd  p ap bðxs Þd n d nþ1  ¼ ðy Þ nþ2  z zn : nþ2 0 ðxs Þp 0 can ap b½ðxs Þd  n p s d 0 ðxs Þp n¼0 0 ca a b½ðx Þ  n¼0 n¼0 ð7Þ

From yz2 ¼ 0, we obtain

x2 ¼ y p z d þ

#

377

1 1 X X nþ1 kþnþ2 yp yn ðzd1 Þ þ zk3 z2 ðzd1 Þ n¼1

!

k¼0

! 1 1 1 X X X 2 nþ2 nþ1 kþnþ2 : þ yp yn ðzd1 Þ þ zk3 z2 ðzd1 Þ ¼ yp zd1 þ z2 ðzd1 Þ þ yp zn3 z2 ðzd1 Þ n¼1

n¼1

ð8Þ

k¼1

Applying aad ðawÞ ¼ aw ¼ ðawÞaad , we get

" yp ¼ 1  yyd ¼ and

" n

p

y y ¼

p  ðawÞd a

#

ðawÞd b

cad ðawÞd a ð1  pÞ  cad ðawÞd b

aad ðawÞn1 ðawÞp a aad ðawÞn1 ðawÞp b cad ðawÞn1 ðawÞp a

# ðn ¼ 1; 2; . . .Þ

cad ðawÞn1 ðawÞp b

Hence, by (8), observe that

" x2 ¼

2

0 ap b½ðxs Þd   ðawÞd bðxs Þd

# p

1  X an ap

0

"

0 ap b½ðxs Þd 

nþ2

#

þy can1 ap 0 0 0 ðxs Þd  cad ðawÞd bðxs Þd n¼1 0" 2 31 # d kþnþ2 1 1 k p s X X 0 0 a a b½ðx Þ  40 5A: þ þ yn yp @ nþ1 s d kþnþ2 Þ  0 ½ðx n¼1 k¼1 0 cak1 ap b½ðxs Þd  0

ð9Þ

Therefore, (7) and (9) imply (5). We can see that the conditions ap bc ¼ 0 and xs c ¼ 0 are equivalent with the following geometrical conditions: 

bcA  aA and cA  ðxs Þ ; where x ¼ fy 2 A : xy ¼ 0g. We give an example to illustrate our results.  p b 2 A relative to the idempotent p. Since 0 0 ad ¼ a ¼ p; ap ¼ 0; xs ¼ 0 ¼ ðxs Þd , ðxs Þp ¼ 1  p and w ¼ p ¼ aw ¼ ðawÞd , by Theorem 2.1, it follows that x 2 Ad and   p b . xd ¼ 0 0

Example 2.1. Let A be a Banach algebra, p 2 A be an idempotent, and let x ¼



The following result is a straightforward application of Theorem 2.1.

Corollary 2.1. Let x be defined as in (1) where a 2 ðpApÞd and the generalized Schur d xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd ; m be defined as in (2) and let w ¼ aad þ ad bca be such that aw 2 ðpApÞd . (i) If ap bc ¼ 0 and xs ¼ 0, then x 2 Ad and

xd ¼ m þ m2



0 cap

 X  1 0 0 þ mnþ1 n1 p 0 a ca n¼2



0 ca

n2

ap b

:

(ii) If ap bc ¼ 0; cap b ¼ 0; aap b ¼ ap bd and xs ¼ 0, then x 2 Ad and

xd ¼ m 1 þ

 1 X mnþ1 n¼0

0 can ap

0 0

! :

complement

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 220 (2013) 374–381

378

Proof. The part (ii) follows from (i) and aap b ¼ ap bðxs þ cad bÞ ¼ 0.

h

 0 ap b m2 ¼ m holds. 0 0 Following the same strategy as in the proof of Theorem 2.1, we obtain a next representation for xd . For the sake of clarity of presentation, the short proof is given. Observe that the part (ii) of Corollary 2.1 recovers [11,], because the equality m 



Theorem 2.2. Let x be defined as in (1) where a 2 ðpApÞd and the generalized Schur d xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd ; m be defined as in (2) and let w ¼ aad þ ad bca be such that aw 2 ðpApÞd . If p

complement

s

bca ¼ 0 and bx ¼ 0; then x 2 Ad and

" d

x ¼

h

0

i2

#

0

ðxs Þd cap  ðxs Þd cad ðawÞd a ðxs Þd  ðxs Þd cad ðawÞd b " # 1 0 0 X h i h i nþ2 nþ2 þ d ðxs Þd can1 ap ða  bca ðawÞd aÞ ðxs Þd can1 ap bðð1  pÞ  cad ðawÞd bÞ n¼1 " # 1 0 0 X h i h i nþ1 nþ1 þ ðxs Þd cad ðawÞn1 ðawÞp a ðxs Þd cad ðawÞn1 ðawÞp b n¼1 " #! 1 0 0 X h ikþnþ2 h ikþnþ2 þ d d ðxs Þd cak1 ap bca ðawÞn1 ðawÞp a ðxs Þd cak1 ap bca ðawÞn1 ðawÞp b k¼1 " #! n   X 1  1 0 0 X p 0 aap ap b h i nþ2  þ mnþ1 : 0 ðxs Þd can ap b 0 ðxs Þp cap xs n¼0 n¼0

ð10Þ

Proof. Let y and z be defined as in (6). Therefore, x ¼ y þ z and yz ¼ 0. By Lemma 1.4, we get y 2 Ad and yd ¼ m. Assume that

 z1 ¼

0

0



0 xs

 ;

z2 ¼

0

0

cap

0



 and z3 ¼

aap 0

 ap b : 0

d

Now z ¼ z1 þ z2 þ z3 ; z1 2 A , z2 2 Anil and z3 2 Aqnil . The equality z3 z2 ¼ 0 and Lemma 1.1 give z2 þ z3 2 Aqnil . Because ðz2 þ z3 Þz1 ¼ 0, by Lemma 1.3 (i), z 2 Ad and

zd ¼ zd1 þ

1 1 X X nþ1 nþ2 ðzd1 Þ ðz2 þ z3 Þzn1 ¼ zd1 þ ðzd1 Þ z2 zn3 : 3 n¼1

n¼0

Applying Lemma 1.2, x 2 Ad and xd ¼ x1 þ x2 , where

x1 ¼

1 X nþ1 ðzd Þ yn yp

and x2 ¼

n¼0

n¼0

Using z2 y ¼ 0, we have

x1 ¼

zd1 yp

þ

2 ðzd1 Þ z2

! 1 1 1 X X X d nþ2 n p d nþ1 d kþnþ2 k þ ðz1 Þ z2 z3 y þ ðz1 Þ þ ðz1 Þ z2 z3 yn yp n¼1

"

1 X nþ1 zp zn ðyd Þ :

n¼1

k¼1

# h i 2 ¼ ðxs Þd cap  ðxs Þd cad ðawÞd a ðxs Þd  ðxs Þd cad ðawÞd b " # " # 1 1 0 0 0 0 X X h i h i h i p nþ2 nþ2 nþ1 y þ þ can ap ðxs Þd can1 ap b ðxs Þd 0 ðxs Þd n¼1 n¼1 " #! 1 0 0 X h ikþnþ2 h ikþnþ2 yn yp : þ s d k p s d ðx Þ ca a ðx Þ cak1 ap b k¼1 0

0

From (11) and

x2 ¼

" #!   X 1 1 0 0 X p 0 nþ1 h inþ2 ðyd Þ ;  zn s d n p s p Þ ca a b 0 ðx 0 ðx Þ n¼0 n¼0

ð11Þ

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 220 (2013) 374–381

379

we conclude that (10) hold. If we assume that c ¼ 0 in Theorem 2.1 and b ¼ 0 Theorem 2.2, then xs ¼ d and we can obtain [3, Theorem 2.3(i)] as a consequence. Recall that [3, Theorem 2.3(i)] is an extension in Banach algebras of results [15, Lemma 2.2] for matrices and [10, Theorem 5.3] for bounded linear operators. Using Theorem 2.2, we can verify the following corollary. Corollary 2.2. Let x be defined as in (1) where a 2 ðpApÞd and the generalized Schur d xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd ; m be defined as in (2) and let w ¼ aad þ ad bca be such that aw 2 ðpApÞd .

complement

p

(i) If bca ¼ 0 and xs ¼ 0, then x 2 Ad and d

x ¼mþ



0 ap b 0



0

2

m þ

" 1 X 0

#

an1 ap b

mnþ1 :

0 can2 ap b

n¼2

p

p

(ii) If bca ¼ 0; cap b ¼ 0; cap a ¼ dca and xs ¼ 0, then x 2 Ad and

!

xd ¼

1þ

 1  X 0 an ap b mnþ1 m: 0 0 n¼0

The item (ii) of Corollary 2.2 covers [11, Theorem 12], by can ap b ¼ 0 for n ¼ 0; 1; 2; . . . Also remark that Corollary 2.1(i) and Corollary 2.2 (i) recover [17, Corollary 3.5] and Lemma 1.4. Now we state the special cases of Theorem 2.1 and Theorem 2.2. Note that, if w ¼ aad , then aw ¼ a2 ad 2 ðpApÞ# and 2 d # ða a Þ ¼ ad . Corollary 2.3. Let x be defined as in (1) xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd , and let bc ¼ 0. If

a 2 ðpApÞd

where

and

the

generalized

Schur

(1) xs c ¼ 0, then x 2 Ad and

  p 1 X p 0 aa x ¼ mnþ1 1 s p 0 ðx Þ cap n¼0 d

ap b

"

n þ

xs

2

0 ap b½ðxs Þd   ad bðxs Þd 0

#

2

ðxs Þd  cðad Þ bðxs Þd

2 3 nþ2 1 X 0 an ap b½ðxs Þd  4 5; þ nþ2 n¼1 0 can1 ap b½ðxs Þd 

(2) xs c ¼ 0 and a 2 ðpApÞ1 , then x 2 Ad and

#   " 1 X 0 0 0 a1 bðxs Þd nþ1 ; þ x ¼ m2 0 ðxs Þp ðxs Þn 0 ðxs Þd  a2 bðxs Þd n¼0 d

s

(3) bx ¼ 0, then x 2 Ad and

" d

x ¼

h

i2

0

#

0 2

ðxs Þd cap  ðxs Þd cad ðxs Þd  ðxs Þd cðad Þ b  p n   p 0 aa ap b mnþ1 þ um1 n¼0 1 ; 0 ðxs Þp cap xs

þ

" 1 X

h

n¼1

ðxs Þd

0



0 inþ2

s

(4) bx ¼ 0 and a 2 ðpApÞ1 , then x 2 Ad and

xd ¼





0

0

ðxs Þd ca1

ðxs Þd  ðxs Þd ca2 b

þ

1  X p n¼0

0 ðxs Þn ðxs Þp

s

(5) bx ¼ 0 and xs c ¼ 0, then x 2 Ad and

xd ¼ m1 þ



0



0

;

s d

0 ðx Þ

(6) xs ¼ 0, then x 2 Ad and xd ¼ m1 ; where

" m1 ¼

2

ad

ðad Þ b 2

cðad Þ

3

cðad Þ b

#

" and

m2 ¼

a1

a2 b

ca2

ca3 b

# :

mnþ1 2 ;

can ap

h

0 inþ2 s d ðx Þ can1 ap b

#

complement

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 220 (2013) 374–381

380

3. Expressions with the generalized Banachiewicz–Schur form In this section, we give new expressions for xd in terms of the generalized Drazin inverse of generalized Schur complement, the generalized Banachiewicz–Schur form (3) and its powers. Theorem 3.1. Let x be defined as in (1) where a 2 ðpApÞd xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd , and let r be defined as in (3). If

ap bc ¼ 0;

p

ap bd ¼ 0;

and

the

generalized

Schur

p

ðxs Þ ca ¼ 0 and abðxs Þ ¼ 0;

complement

ð12Þ

then x 2 Ad and

xd ¼ r 1 þ r



0 cap

 X  1 0 0 þ rn n1 p 0 a ca n¼2

!

0

ð13Þ

:

can2 ap b

Proof. For

" y¼

a2 ad

aad b

caad

d

#

 and

z¼

we have x ¼ y þ z and, by (12), zy ¼ 0. Set Ay  a2 ad ; By  aad b; C y  caad

aap

ap b

cap

0

 ð14Þ

;

p s Ay 2 ðpApÞ# ; A# ¼ ad , ys ¼ Dy  C y A# y By ¼ x ; Ay By ¼  y 0 0 2 Aqnil implying y 2 Ad and yd ¼ r, by C y Apy ¼ 0; By ðys Þp ¼ ad ðabðxs Þp Þ ¼ 0, ðys Þp C y ¼ ððxs Þp caÞad ¼ 0 and Y y ¼ 0 xs ðxs Þp Lemma 1.5. Assume that

z1 ¼



aap

ap b

0

0



and z2 ¼



and

0

0

cap

0

Dy  d.

Then

 :

qnil

Now z1 z2 ¼ 0; z1 2 A and z22 ¼ 0. Using Lemma 1.1, we deduce that z ¼ z1 þ z2 2 Aqnil . P P nþ1 n nþ1 n Applying Lemma 1.3 (i), x 2 Ad and xd ¼ 1 z ¼ r þ r2 z þ 1 z . Since, for n ¼ 2; 3; . . ., n¼0 r n¼2 r

"

n

z ¼

an1 ap ¼z 0

zzn1 1

rz ¼ rz1 and rzn ¼ r



an2 ap b 0

0 can1 ap

#

"

¼ 

an ap can1 ap

# an1 ap b ; can2 ap b

0 , we get (13). can2 ap b

h

A geometrical reformulation of conditions (12) is as follows:

bcA  aA;

bdA  aA;





caA  xs A and ðxs Þ  ðabÞ :

By Theorem 3.1, we can verify the following corollary. Corollary 3.1. Let x be defined as in (1) where a 2 ðpApÞd xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd , and let r be defined as in (3).

and

the

generalized

(i) If equalities (12), aap b ¼ 0 and cap b ¼ 0 hold, then x 2 Ad and

xd ¼ r 1 þ

 1 X r nþ1 n¼0

0

0

can ap

0

! ;

(ii) If ap b ¼ 0 and xs is invertible, then x 2 Ad and

xd ¼ r 1 1 þ

 1 X r nþ1 1 n¼0

0

0

can ap

0

! ;

(iii) If a is invertible, ðxs Þp c ¼ 0 and bðxs Þp ¼ 0, then x 2 Ad and x ¼ r 2 ; (iv) If a and xs are invertible, then x 2 Ad and x ¼ r2 ; where

" r1 ¼

ad þ ad bðxs Þ1 cad

ad bðxs Þ1

ðxs Þ1 cad

ðxs Þ1

#

" and r 2 ¼

a1 þ a1 bðxs Þ1 ca1

a1 bðxs Þ1

ðxs Þ1 ca1

ðxs Þ1

# :

Schur

complement

D. Mosic´, D.S. Djordjevic´ / Applied Mathematics and Computation 220 (2013) 374–381

381

Similarly to Theorem 3.1, we can show the following theorem. Theorem 3.2. Let x be defined as in (1) where a 2 ðpApÞd xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd , and let r be defined as in (3). If p

bca ¼ 0;

p

dca ¼ 0;

p

and

the

generalized

Schur

p

ðxs Þ ca ¼ 0 and abðxs Þ ¼ 0;

complement

ð15Þ

d

then x 2 A and d

x ¼

1þ



" # !  1 X 0 an1 ap b n r r: rþ 0 can2 ap b n¼2 0

0 ap b 0

ð16Þ

Proof. Define y and z as in (14). Then yz ¼ 0 and, in the analogy way as in the proof of Theorem 3.1, we prove the formula (16). h Also we can check the next result using Theorem 3.2. Corollary 3.2. Let x be defined as in (1) where a 2 ðpApÞd and the generalized xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd ; r be defined as in (3), and let r 1 be defined as in Corollary 3.1.

Schur

complement

(i) If equalities (15), caap ¼ 0 and cap b ¼ 0 hold, then x 2 Ad and

xd ¼

1þ

!  1  X 0 an ap b nþ1 r: r 0 0 n¼0

(ii) If cap ¼ 0 and xs is invertible, then x 2 Ad and d

x ¼

!  1  X 0 an ap b nþ1 r1 1þ r1 : 0 0 n¼0

From Corollary 3.1 (ii) or Corollary 3.2 (ii), we get an extension of result for the Drazin inverse of a block matrix by Wei [23]. Corollary 3.3. Let x be defined as in (1) where a 2 ðpApÞd and the generalized Schur complement xs ¼ d  cad b 2 ðð1  pÞAð1  pÞÞd , and let r 1 be defined as in Corollary 3.1. If ap b ¼ 0; cap ¼ 0 and xs is invertible, then x 2 Ad and x ¼ r1 . 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]

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