Perturbation bounds for the generalized inverse AT,S(2) with respect to the Frobenius norm

Applied Mathematics and Computation 218 (2011) 4595–4604

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

ð2Þ

Perturbation bounds for the generalized inverse AT;S with respect to the Frobenius norm q Xiaoji Liu a,⇑, Yonghui Qin a, Yaoming Yu b a b

College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, PR China School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia

a r t i c l e

i n f o

Keywords: Perturbation bound ð2Þ Generalized inverse AT;S inverse Frobenius norm

a b s t r a c t In this paper, we present the perturbation bounds and the relative error bound of the genð2Þ eralized inverse AT;S with respect to the Frobenius norm, and apply the results to the relative errors of the solution of the general restricted linear equation. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction ð2Þ

In recent years, the perturbation bounds of the generalized inverse AT;S of operators or matrices have been widely investigated (see, for example, [11,10,14]). In [11], the authors put forward a certain condition, named (W) condition, and disð2Þ cussed the perturbation bounds of AT;S under such a condition. In [10], the authors presented lower and upper bounds for ð2Þ ð2Þ the perturbation of AT;S under the condition kEAT;S k2 < 1, where a perturbation E of A also satisfies some conditions. In ð2Þ [14], the authors derived the perturbation bounds of AT;S with respect to the spectral norm when the subspaces T and S also have perturbation, and they also discussed the applications to the constrained linear system. In [13] the authors considered the perturbation problem for the bounded of outer inverse and the bounded generalized inverse of linear operators with respect to the projectors, and gave new perturbation theorems by the generalized Neumman lemma in Banach spaces. In [7], the authors presented the perturbation bounds of the Moore–Penrose inverse under the Frobenius norm by using the sinð2Þ gular value decomposition. To our certain knowledge, the perturbation bound of the generalized inverse AT;S of a matrix A with respect to the Frobenius norm have not been yet considered. ð2Þ In this paper, we will investigate the perturbation bound of the generalized inverse AT;S with respect to the Frobenius norm. In the section, we will introduce some notation and lemmas. In Section 2, we will present mainly the perturbation ð2Þ bound of AT;S with respect to the Frobenius norm when subspaces have also perturbation. In Section 3, we will discuss ð2Þ ð2Þ the perturbation bound of AT;S under a certain condition, which generalizes kEAT;S k < 1. In Section 4, we apply our results to the relative errors of the solution of the general restricted linear equation. In Section 5, we will take an example to illustrate our results. Throughout this paper, the symbol Cmn denotes a set of all m  n complex matrices. Let A 2 Cmn , the symbols RðAÞ; N ðAÞ and rank (A) stand for its range, null space and rank, respectively. If L1 is a subspace of L2, then we write L1  L2. The symbols k  kF, k  k and k  k2 denote the Frobenius norm (F-norm for short), the unitarily invariant norm and the spectral norm of a matrix, respectively. Note that kABk 6 kAk2kBk, kAkkBk2 (see [8, Theorem 2.3.9]) and kAk2 6 kAk (see [9, Theorem 2.3.5]).

q This work was supported by the National Natural Science Foundation of China (11061005), the Ministry of Education Science and Technology Key Project under Grant 210164 and Grant (HCIC201103) of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund. ⇑ Corresponding author. E-mail addresses: [email protected] (X. Liu), [email protected] (Y. Qin), [email protected] (Y. Yu).

0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.043

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X. Liu et al. / Applied Mathematics and Computation 218 (2011) 4595–4604

Lemma 1.1. [2, Lemma 2.1]. Let U ¼ ðU 1 ; U 2 Þ 2 Cmm and V ¼ ðV 1 ; V 2 Þ 2 Cnn be both unitary matrices. Then

kEk2F ¼ kU 1 EV 1 k2F þ kU 1 EV 2 k2F þ kU 2 EV 1 k2F þ kU 2 EV 2 k2F ; mr

ns

ð1:1Þ

mn

where U 1 2 C ; V 1 2 C and E 2 C . Using the CS decomposition ([8, Theorem 2.5.1]), we can easily obtain the following result, which appeared and was proved in an alternative way in [3]. Lemma 1.2. Let U 2 Cnn be a unitary matrix with the form

 U¼

U 11

U 12

U 21

U 22

 ;

where U 11 2 Ckk ; 0 < k < n. Then kU12k = kU21k. The following result is the matrix version of [6, Lemma 2.2] and the condition k1 + k2 > 0, which is necessary to (1.3) and nonzero P in Lemma 1.3, is added to it. Lemma 1.3. Let P 2 Cnn be such that

kPxk 6 k1 kxk þ k2 kðI þ PÞxk;

8x 2 C n ;

ð1:2Þ

where k1 < 1, k2 < 1 and k1 + k2 > 0. Then k1,k2 2 (1, 1) and (I + P)1 exists. Moreover,

1  k1 1 þ k1 kxk 6 kðI þ PÞxk 6 kxk; 8x 2 Cn ; 1 þ k2 1  k2 1  k2 1 þ k2 kyk 6 kðI þ PÞ1 yk 6 kyk; 8y 2 Cm : 1 þ k1 1  k1

ð1:3Þ ð1:4Þ

A matrix B is called a {2}-inverse of matrix A if BAB = B holds. The symbols A , Ind (A) and AD denote, respectively, the Moore– Penrose inverse, the index and the Drazin inverse of A, and, obviously, rank (A ) = rank(A) (see [1] for details). Lemma 1.4 [1, Theorem 2.14]. Let A 2 Cmn with rank r, let T be a subspace of Cn of dimension s 6 r, and let S be a subspace of Cm of dimension m  s. Then, A has a {2}-inverse X such that RðXÞ ¼ T and N ðXÞ ¼ S if and only if

AT  S ¼ Cm ; in which case X is unique. ð2Þ Such an X in Lemma 1.4 is called the generalized inverse of A with prescribed range T and null space S and denoted by AT;S . And ð2Þ

AT;S ¼ ðPS? AP T Þy ;

ð1:5Þ

where the symbol PL stands for the orthogonal projector on the subspace L ([1, Ex. 2.61]). The following lemma is gained by simplifying [5, Theorem 2.2]. ð2Þ

Lemma 1.5. Let A; B; E 2 Cmn with B = A + E, and T and S be subspaces of Cn and Cm , respectively. Suppose that AT;S exists. Then the following statements are equivalent. ð2Þ

(i) BT;S exists. ð2Þ

(ii) In þ AT;S E is invertible. ð2Þ

(iii) Im þ EAT;S is invertible.In which case,

 1  1 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ BT;S ¼ In þ AT;S E AT;S ¼ AT;S Im þ EAT;S :

ð1:6Þ

2. General situation ð2Þ

In this section, we will discuss the perturbation bounds for the generalized inverse AT;S . The following is our main result in which the subspaces T and S also have perturbation. e  Cn and S; e Theorem 2.1. Let A; B; E 2 Cmn with B = A + E, and let T; T S  Cm . If AT;S and B exist, then eT ;eS ð2Þ

ð2Þ

X. Liu et al. / Applied Mathematics and Computation 218 (2011) 4595–4604 ð2Þ

4597

ð2Þ

kB  AT;S kF eT ;eS   ð2Þ ð2Þ 6 max kAT;S k22 ; kB k22 kDkF eT ;eS  h   i ð2Þ ð2Þ 6 max kAT;S k22 ; kB k22 kEkF þ kPe?  PS? kF þ kPe  PT kF kAkF ; eT ;eS T S

ð2:1Þ ð2:2Þ

D ¼ Pe? BPe  P S? APT . T

S

 y ð2Þ ð2Þ Proof. By (1.5), B ¼ Pe? BPe and AT;S ¼ ðP S? APT Þy . So the singular value decomposition is made to PS? AP T and Pe? BPe, eT ;eS T T S S respectively, as follows

 PS? AP T ¼ U

R1 0 0

0

 V

 e R2 and Pe? BPe ¼ U T S 0

0



0

e ; V

ð2:3Þ

where

R1 ¼ diagðr1 ; r2 ; . . . ; rr Þ;

r1 P r2 P    P rr > 0; ~ 1; r ~ 2; . . . ; r ~ s Þ; r ~1 P r ~2 P    P r ~ s > 0; R2 ¼ diagðr e ¼ ðU e 1; U e 2 Þ 2 Cmm and V ¼ ðV 1 ; V 2 Þ; V e ¼ ðV e 1; V e 2 Þ 2 Cnn all are r ¼ rankðP S? APT Þ and s ¼ rankðP S? BP T Þ, and U ¼ ðU 1 ; U 2 Þ; U mr ms nr ns e e unitary matrices, where U 1 2 C ; U 1 2 C ; V 1 2 C ; V 1 2 C . So

e e 1 R2 V and Pe? BPe ¼ U 1 T S

ð2:4Þ

e   U 1 R1 V  ; e 1 R2 V D ¼ Pe? BPe  PS? APT ¼ U 1 1

ð2:5Þ

PS? AP T ¼ U 1 R1 V 1 and S

ð2Þ

AT;S ¼ V

T

R1 0 1 0

!

!

ð2Þ e B ¼V eT ;eS

U ;

0

R1 0 e 2 U : 0

ð2:6Þ

0

Thus, by (2.6) and (2.5),

!

e  ðBð2Þ  Að2Þ ÞU ¼ V T;S eT ;eS

R1 0 2

¼

0

0

!

1 R1 0 e 0 2 e  V R1 U UV

0

e  U1 U 1 e  U1 U

0

e  U2 U 1 e  U2 U

2

0

! 

2

eV1 V 1 eV1 V

1 e e e R1 R1 2 U 1 U 1  V 1 V 1 R1 2 U 1 U2

¼

e  V 1 R1 V 1 2

0

e  V2 V 1 e  V2 V

2

!

!

R1 0 1 0

2

0

! ð2:7Þ

0

0

1 e R1 2 U 1 DV 1 R 1  ¼@   e 2 R2 U1 DV 1

!

 e  R2 2 ðU 2 D V 1 Þ

0

1 A:

ð2:8Þ

Similarly,

  ð2Þ ð2Þ e ¼ V  B  AT;S U eT ;eS

e 1 R1  R1 U  U e V 1 V 2 1 1 1 e 1 V V 1R 2

 e 1 R1 1 U 1 D V 1 R2 B ¼@   e  DV 2 R2 U 2

! ð2:9Þ

0

2

0

1

e R1 1 U1 U 2



e  DV 1 R2 U 1 2 0

 1 C A:

ð2:10Þ

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By (2.8) and Lemma 1.1, we obtain

       1  e ð2Þ ð2Þ e 1 k2 þ 1 k U  D V e 1 k2 þ k U e  DV 2 k 2 2kB  AT;S k2F 6 2 2 k U DV 1 k2F þ kU 1 D V 1 1 2 F F F eT ;eS rr r~ s r~ 4s         1 e 2 k2 þ k U e  DV 1 k2 þ 4 k U 1 D V F F 2 rr    1 1 e 1 k2 þ kU  D V e 2 k2 þ kU  D V e 1 k2 e  DV 1 k 2 þ k U e  DV 2 k2 þ k U e  DV 1 k2 þ kU  D V kU 6 max ; 1 1 2 1 1 2 F F F F F F r4r r~ 4s   ð2Þ ð2Þ 6 2 max kAT;S k42 ; kB k42 kDk2F : eT ;eS Hence (2.1) holds. Since

  D ¼ Pe? EPe þ Pe?  PS? APe þ PS? AðPe  P T Þ T T T S S and kP S? k2 ¼ kPe? k2 ¼ kPe k2 ¼ 1, by (2.1) and (2.2) is reached. h T

S

ð2Þ

Remark 2.1. In [15, Theorem 2.5], authors considered the perturbation of AT;S along with a perturbation N ðHÞ of the null space S and presented its expression

h i1    1  1 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ðA þ EÞT;NðHÞ ¼ I þ AT;S E AT;S þ I þ AT;S E AT;S  I þ ðAGÞg AFAAT;S ðAGÞg AF I  AAT;S ; where (AG)g denotes the group inverse of AG and G satisfies RðGÞ ¼ T and N ðGÞ ¼ S. By the hypotheses in [15, Theorem 2.5] and (1.6), it is not difficult to deduce its perturbation bound with respect to Fnorm as follows.

 1  1 h ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ kðA þ EÞT;NðHÞ  AT;S kF 6 kAT;S kF k I þ EAT;S kF k I þ ðAGÞg AFAAT;S kF kðAGÞg AFkF þ kðAGÞg AFAAT;S kF i ð2Þ ð2Þ þ kðAGÞg AFEAT;S kF þ kEAT;S kF :

ð2:11Þ

e ¼ T. Let Here is an example for showing that our perturbation bound g1 in (2.1) is not worse than that g2 in (2.11) when T

3

2

62 0 1 0 07 7 6 A¼6 7; 40 1 2 0 05

6 6 E¼6 4

2

4 2 1 1 0

0 2

0

0

1 0 0

6 60 1 0 6 G¼6 60 0 1 6 40 0 0 0

0

0

0 0 3 0 7 07 7 07 7: 7 05 0

0:001 0 0 0

0

0

0 0

3

0:001 0:003 0 0 7 7 7; 0 0 0 05 0

0

0 0

2

0

0:0002 0:0001

6 0 6 0 6 F¼6 0:0003 0:0002 6 6 0:0001 4 0 0

0

0 0 0 0

0

3

7 0:0006 7 7 7; 0 7 7 0 5 0

Then from (2.1) and (2.11), we have

g1 ¼ 0:00349969327173 < g2 ¼ 0:01244198321672: ð2Þ ð2Þ e ¼ RðB Þ; e Taking T ¼ RðA Þ; S ¼ N ðA Þ and T S ¼ N ðB Þ in Theorem 2.1 yields Ay ¼ AT;S and By ¼ B , respectively, and eT ;eS

BBy ¼ PRðBBy Þ ¼ PRðBÞ ¼ PN ðB Þ? ¼ Pe? ; S By B ¼ PRðBy BÞ ¼ PRðB Þ ¼ Pe; T AAy ¼ P S? ;

Ay A ¼ P T :

So D ¼ Pe? BPe  P S? AP T ¼ B  A ¼ E and then we have the following corollary. S

T

Corollary 2.1 [7, Theorem 2.1]. Let A; B; E 2 Cmn with B = A + E. Then

n o kBy  Ay kF 6 max kAy k22 ; kBy k22 kEkF

ð2:12Þ

ð2Þ e ¼ RðBk2 Þ; e Taking T ¼ RðAk1 Þ; S ¼ N ðAk1 Þ and T S ¼ N ðBk2 Þ in Theorem 2.1, where k1 = Ind(A) and k2 = Ind(B), yields AD ¼ AT;S ð2Þ D and B ¼ B , respectively. So we have the perturbation bound of the Drazin inverse. eT ;eS

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Corollary 2.2. Let A; B; E 2 Cmm with B = A + E. Then

n o kBD  AD kF 6 max kAD k22 ; kBD k22 kDkF

ð2:13Þ

where D is the same as Theorem 2.1. e; S ¼ e In the following, if we only consider the perturbation of A, i.e., T ¼ T S, we have the following results. ð2Þ

ð2Þ

Corollary 2.3. Let A; B; E 2 Cmn with B = A + E, and let T  Cn and S  Cm . If AT;S and BT;S exist, then ð2Þ

ð2Þ

ð2Þ

ð2Þ

kBT;S  AT;S kF 6 kAT;S k2 kBT;S k2 kP S? EPT kF 6

ð2:14Þ

ð2Þ ð2Þ kAT;S k2 kBT;S k2 kEkF :

ð2:15Þ

     1 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ Proof. By Lemma 1.5,BT;S ¼ I þ AT;S E AT;S and then rank BT;S ¼ rank AT;S . By (1.5), BT;S ¼ ðPS? BP T Þy and AT;S ¼ ðPS? AP T Þy and therefore rankðP S? BP T Þ ¼ rankðP S? APT Þ. So we have also (2.3) where r = s and

D ¼ PS? BPT  P S? APT ¼ PS? EP T :

ð2:16Þ

e  U is unitary, by Lemma 1.2 and (2.16), Since U

e  U2 k ¼ kU e  U1 k ¼ kU e  DV 1 R1 k 6 k U e  DV 1 k kR1 k ¼ 1 k U e  DV 1 k 2 : kU F F F F 2 1 1 1 2 2 2 2 F

ð2:17Þ

rr

Analogically, we have

1  e  e 1 e 2 k ¼ kU  U e kU 1 U F 2 1 kF ¼ kU 2 D V 1 R2 kF 6 ~ kU 2 D V 1 kF ; rr e  DV 2 k 6 1 k U e  V 2 k ¼ kR1 U e  DV k ; e  V 1k ¼ kV kV F F F 2 2 1 1 r~ r 1 2 F 1 1  e e 1 k ¼ kV  V e e 2k : kV 2 V kU 1 D V F F 1 2 kF ¼ kR1 U 1 D V 2 kF 6

ð2:18Þ ð2:19Þ ð2:20Þ

rr

By (2.7), (2.9) and (2.17)–(2.20), and Lemma 1.1, we obtain

     1  e e 1 k2 þ 1 k U e 1 k2 þ 1 kU  U e 2 k2 þ k V e  U 2 k2 þ kV  V e  V 1 k2 k U 1 DV 1 k2F þ kU 1 D V 2 1 1 2 F F F F F 2 2 rr r~ r rr  1  e e 1 k2 þ k U e 2 k2 þ kU  D V e 1 k2 þ k U e  DV 1 k2 þ kU  D V e  DV 2 k 2 6 2 2 k U 1 DV 1 k2F þ kU 1 D V 2 1 F F F F F 1 2 rr r~ r

2kBT;S  AT;S k2F 6 ð2Þ

ð2Þ

2 ~2 r r

6 2kAT;S k22 kBT;S k22 kDk2F : ð2Þ

ð2Þ

Hence (2.14) holds and then (2.15) is easily reached. h Note that rank(A) = rank(B) implies RðA Þ ¼ RðB Þ and N ðA Þ ¼ N ðB Þ. So, by (2.15), we have the following result. Corollary 2.4 [7, Theorem 2.2]. Let A; B; E 2 Cmn with B = A + E. If rank(A) = rank(B), then

kBy  Ay kF 6 kAy k2 kBy k2 kEkF : Similar to the argument above Corollary 2.4, we have the perturbation bound of the Drazin inverse. Corollary 2.5. Let A; B; E 2 Cmm with B = A + E. If rank(Ak) = rank(Bk), where k = max{Ind(A), Ind(B)}, then

kBD  AD kF 6 kAD k2 kBD k2 kEkF :

ð2:21Þ

3. Special situation ð2Þ

In a great deal of cases of the perturbation bound, we do not know beforehand whether BT;S exists. So we often judge the ð2Þ ð2Þ ð2Þ existence of BT;S based on the situations of AT;S and E. The following is such a case, in which EAT;S satisfies (1.2) in Lemma 1.3. ð2Þ

Theorem 3.1. Let A; B; E 2 Cmn with B = A + E and let T  Cn and S  Cm . Assume that AT;S exists and satisfies ð2Þ

ð2Þ

kEAT;S yk 6 k1 kyk þ k2 kðI þ EAT;S Þyk;

8y 2 Cm ; ð2Þ

ð3:1Þ ð2Þ

ð2Þ



ð2Þ

where k1,k2 2 (1,1) and k1 + k2 > 0. Then BT;S exists and BT;S ¼ AT;S I þ EAT;S

1

.

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Furthermore ð2Þ

ð2Þ

ð2Þ

ð2Þ

kBT;S  AT;S kF 6 kBT;S  AT;S kF ð2Þ kAT;S kF

6

1 þ k2 ð2Þ 2 kA k kP ? EPT kF ; 1  k1 T;S 2 S

ð3:2Þ

1 þ k2 ð2Þ kA k kP ? EP T kF : 1  k1 T;S 2 S

ð3:3Þ

 1 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ Proof. By (3.1) and Lemma 1.3, I þ EAT;S is invertible and therefore, by Lemma 1.5, BT;S exists and BT;S ¼ AT;S I þ EAT;S . By Lemma 1.3, we have

 1 1  k2 1 þ k2 ð2Þ 6 k I þ EAT;S k2 6 : 1 þ k1 1  k1 So

 1 1 þ k2 ð2Þ ð2Þ ð2Þ ð2Þ kBT;S k2 ¼ kAT;S I þ EAT;S  k2 6 kA k : 1  k1 T;S 2

ð3:4Þ

Obviously D ¼ PS? EP T in Theorem 2.1. So by (2.14) and (3.4), we obtain (3.2). ð2Þ ð2Þ From kAT;S k2 6 kAT;S kF , (3.3) follows. h Remark 3.1. In [12], authors introduced PQ–norm k  kPQ of matrices, kAkQP = kQAP1k2 andkykQ = kQyk2 where P 2 Cnn and ð2Þ Q 2 Cmm are nonsingular, and discussed the perturbation bound [12, Theorem 4.1] for AT;S . From Theorem 3.1, we can get a result similar to their Theorem 4.1. We explain this in detail as follows. ð2Þ Let X ¼ PAT;S Q 1 . Then it is easy to verify that XðQAP1 ÞX ¼ X; RðXÞ ¼ PT and N ðXÞ ¼ QS. Therefore ð2Þ

ð2Þ

ð2Þ

ð2Þ

ðQAP1 ÞPT;QS ¼ X ¼ PAT;S Q 1 . Now, take Q  norm kykQ in (3.1). If kAT;S kPQ kEkQP < 1, then take k1 ¼ kAT;S kPQ kEkQP and k2 = 0. Thus ð2Þ

ð2Þ

kEAT;S ykQ 6 kEkQP kAT;S kPQ kykQ ¼ k1 kykQ def

and therefore (3.1) holds. Note that kAkQP ¼ kQAP1 k2 6 kQAP1 kF ¼ kAkFQP . Thus ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

ð2Þ

kðA þ EÞT;S  AT;S kPQ ¼ kPðA þ EÞT;S Q 1  PAT;S Q 1 k2 6 kðQ ðA þ EÞP1 ÞPT;QS  ðQAP1 ÞPT;QS kF 1 1 ð2Þ ð2Þ kðQAP1 ÞPT;QS k22 kPðQSÞ? QEP1 PPT kF 6 kA k2 kP ? k kQEP1 kF kPPT k2 1  k1 1  k1 T;S PQ ðQSÞ 2 KT;S ðAÞkEkQP =kAkQP ð2Þ ¼l kA k ; 1  KT;S ðAÞkEkQP =kAkQP T;S PQ 6

where

l ¼ kEkFQP =kEkQP and KT;S ðAÞ ¼ kAkQP kAð2Þ T;S kPQ . As a result, ð2Þ

ð2Þ

kðA þ EÞT;S  AT;S kPQ ð2Þ kAT;S kPQ

6l

KT;S ðAÞkEkQP =kAkQP : 1  KT;S ðAÞkEkQP =kAkQP

When l = 1, we can get [12, Theorem 4.1]. ð2Þ ð2Þ In literature one often supposes kEAT;S k < 1 when discussing the perturbation of the generalized inverse AT;S . In Theorem ð2Þ

ð2Þ

ð2Þ

3.1, the condition (3.1) generalizes kEAT;S k < 1. Indeed, if kEAT;S k < 1, then, taking k1 ¼ kEAT;S k and k2 = 0, we have

  ð2Þ ð2Þ ð2Þ kEAT;S yk 6 kEAT;S kkyk 6 k1 kyk þ k2 k I þ EAT;S yk for any kyk. We therefore have the following corollary from Theorem 3.1. ð2Þ

ð2Þ

Corollary 3.1. Let A; B; E 2 Cmn with B = A + E and let T  Cn and S  Cm . If kEAT;S kF < 1, then BT;S exists and

kAT;S k22 ð2Þ

ð2Þ

ð2Þ

kBT;S  AT;S kF 6 ð2Þ

ð2Þ

kBT;S  AT;S kF ð2Þ kAT;S kF

ð2Þ

1  kEAT;S k2

kPS? EP T kF ;

ð3:5Þ

kPS? EP T kF :

ð3:6Þ

ð2Þ

6

kAT;S k2 ð2Þ

1  kEAT;S k2

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Proof. Since kEAT;S k < 1, by the argument above and (3.2) and (3.3), we have the two inequations. h ð2Þ

If AT;S satisfies the condition below ð2Þ

ð2Þ

kAT;S Eyk 6 k1 kyk þ k2 kðI þ AT;S EÞyk;

8y 2 Cn

ð3:7Þ

where k1, k2 2 (1, 1) and k1 + k2 > 0, then the following result, similar to the proof of Theorem 3.1, holds. ð2Þ

Theorem 3.2. Let A; B; E 2 Cmn with B = A + E and let T  Cn and S  Cm . Assume that AT;S exists and satisfies (3.7). Then  1 ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ AT;S . Furthermore I þ AT;S E is invertible and BT;S exists and BT;S ¼ I þ AT;S E ð2Þ

ð2Þ

kBT;S  AT;S kF 6 ð2Þ kBT;S

ð2Þ  AT;S kF ð2Þ kAT;S kF

6

1 þ k2 ð2Þ 2 kA k kP ? EP T kF ; 1  k1 T;S 2 S

ð3:8Þ

1 þ k2 ð2Þ kA k kP ? EP T kF : 1  k1 T;S 2 S

ð3:9Þ

ð2Þ

ð2Þ

Corollary 3.2. Let A; B; E 2 Cmn with B = A + E and let T  Cn and S  Cm . If kEAT;S kF < 1, then BT;S exists and ð2Þ

ð2Þ

kBT;S  AT;S kF 6 ð2Þ

1

ð2Þ kAT;S k22 ð2Þ  kEAT;S k2

ð2Þ

kBT;S  AT;S kF ð2Þ kAT;S kF

kPS? EP T kF ;

ð3:10Þ

kPS? EP T kF :

ð3:11Þ

ð2Þ

kAT;S k2

6

ð2Þ

1  kEAT;S k2

In the paper [6,7] the multiplicative perturbation model has been considered. In fact, it is a special additive perturbation ð2Þ model. In the following, we consider the multiplicative perturbation of the generalized inverse AT;S : Corollary 3.3. Let A 2 Cmn and B ¼ D1 AD2 , where D1 2 Cm and D2 2 Cn are invertible, with E = B  A. Let T  Cn and S  Cm . ð2Þ ð2Þ Assume that AT;S exists and satisfies (3.1) or (3.7). Then BT;S exists and

kBT;S  AT;S kF 6 wkAT;S k22 ; ð2Þ

ð2Þ

ð2Þ kBT;S

ð2Þ  AT;S kF ð2Þ kAT;S kF

ð2Þ

ð3:12Þ

ð2Þ

ð3:13Þ

6 wkAT;S k2 ;

 2 where w ¼ 1þk kP S? ðIm  D1 ÞAP T kF þ kPS? D1 AðIn  D2 ÞP T kF . 1k1 Proof. From the equation

 E ¼ B  A ¼ D1 AD2  A ¼  Im  D1 A  D1 AðIn  D2 Þ; we can easily show the corollary. h ð2Þ

We end this section with the following result, concerning the perturbation bound of the oblique projection AAT;S . ð2Þ

Theorem 3.3. Let A; B; E 2 Cmn with B = A + E and let T  Cn and S  Cm . Assume that AT;S exists and satisfies (3.1) or (3.7). Then ð2Þ BT;S exists and ð2Þ

ð2Þ

kBBT;S  AAT;S kF 6 where

1 þ k2 ð2Þ ½jðAÞ þ 1kAT;S k2 kEkF ; 1  k1

ð3:14Þ

jðAÞ ¼ kAk2 kAð2Þ T;S k2 . ð2Þ

Proof. We will only prove the case of AT;S satisfying (3.1). The proof of the other case is similar. ð2Þ By Theorem 3.1, BT;S exists. By (3.2) and (3.4), we have

  ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ kBBT;S  AAT;S kF ¼ kA BT;S  AT;S  EBT;S kF 6 kAk2 kBT;S  AT;S kF þ kEkF kBT;S k2 i 1þk 1 þ k2 h 2 ð2Þ ð2Þ ð2Þ 6 kAk2 kAT;S k22 kPS? EP T kF þ kEkF kAT;S k2 6 ½jðAÞ þ 1kEkF kAT;S k2 : 1  k1 1  k1



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4. Application In this section, we will study the relative error of the solution of the general restricted linear equation

Ax ¼ b;

x 2 T;

ð4:1Þ

where A 2 Cmn and b 2 RðAÞ. To this end, we need the following result. Lemma 4.1 [4, Theorem 2.2]. Let A 2 Cmn ; T  Cn and the condition

b 2 AT

and dimAT ¼ dimT ð2Þ

be satisfied. Then the unique solution of (4.1) is given by x ¼ AT;S b, for any subspace S of Cm satisfying AT  S ¼ Cm . Let E and f be perturbed matrix and vector of A and b, respectively. So we have the general restricted linear equation

x 2 Te ;

ðA þ EÞx ¼ b þ f ;

ð4:2Þ

where A; E 2 Cmn and b þ f 2 RðA þ EÞ. From the above argument we have following result. e  Cn and S; e Theorem 4.1. Let A; E 2 Cmn and let T; T S  Cm . Assume that AT;S and ðA þ EÞ exist. If b 2 AT and e T ;e S e , then b þ f 2 ðA þ EÞ T ð2Þ

ð2Þ

  ky  xk2 kf k2 ð2Þ ð2Þ ð2Þ 6 max kAT;S k22 ; kðA þ EÞ k22 kDkF kAkF þ kðA þ EÞ kF kAkF eT ;eS eT ;eS kxk2 kbk2

ð4:3Þ

where D ¼ Pe? ðA þ EÞPe  P S? APT . T

S

e ¼ dim T e . So, by Lemma 4.1, (4.2) has the unique Proof. Since ðA þ EÞ exists, by Lemma 1.4 and [4, Lemma 2.1], dimðA þ EÞ T eT ;eS ð2Þ ð2Þ e . In like manner, x ¼ A b is the unique solution of (4.1). Note that solution y ¼ ðA þ EÞ ðb þ f Þ because b þ f 2 ðA þ EÞ T T;S eT ;eS kbk2 = kAxk2 6 kAkFkxk2. So by (2.1), we obtain ð2Þ

  h i ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ky  xk2 ¼ k ðA þ EÞT;S  AT;S b þ ðA þ EÞT;S f k2 6 max kAT;S k22 ; kðA þ EÞ k22 kDkF kAkF kxk2 þ kðA þ EÞ kF kf k2 eT ;eS eT ;eS   kf k2 ð2Þ ð2Þ ð2Þ 6 max kAT;S k22 ; kðA þ EÞ k22 kDkF kAkF kxk2 þ kðA þ EÞ kF kAkF kxk2 : eT ;eS eT ;eS kbk2 Hence we reach (4.3).

h

ð2Þ

If AT;S and E satisfy (3.1), then we have the result below. ð2Þ

Theorem 4.2. Let A; E 2 Cmn and let T  Cn and S  Cm . Assume that AT;S exists and satisfies (3.1). If b 2 AT and b + f 2 (A + E)T, then

  ky  xkF 1 þ k2 kf k2 ð2Þ ; 6 jðAÞ kAT;S k2 kPS? EP T kF þ kxkF 1  k1 kbk2 where

ð4:4Þ

jðAÞ ¼ kAð2Þ T;S kF kAkF . ð2Þ

ð2Þ

ð2Þ

Proof. By Theorem 3.1, ðA þ EÞT;S exists. Similar to the proof of Theorem 4.1, we have that y ¼ ðA þ EÞT;S ðb þ f Þ and x ¼ AT;S b ð2Þ ð2Þ are the unique solutions of (4.2) and of (4.1), respectively. So by (3.8) and kAT;S k2 6 kAT;S kF , we can get ð2Þ

ð2Þ

ð2Þ

ky  xk2 6 kðA þ EÞT;S  AT;S kF kbk þ kðA þ EÞT;S kF kf k2 6 6

  1 þ k2 kf k2 ð2Þ kxk2 : jðAÞ kAT;S k2 kPS? EP T kF þ 1  k1 kbk2

Hence we reach (4.4).

h

  1 þ k2 kf k2 ð2Þ ð2Þ kAT;S k22 kPS? EP T kF kAkF kxk2 þ kAT;S kF kAkF kxk2 1  k1 kbk2

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5. Example ð2Þ

Here is an example for illustrating the perturbation bounds (3.2) of the generalized inverse AT;S being affected by k1 and k2. When taking appropriately k1 and k2, we can obtain better perturbation bounds. Example 5.1. Let

0

5:3601

4:0232

3:6913

6:5097

2:3395

18:9146

1

C B B 6:7872 3:7518 5:8766 2:0152 7:5335 1:3783 C C B C B 6:2610 1:7889 17:4413 C B 8:9443 13:4164 13:4164 C B C B A ¼ B 10:7105 18:2179 19:6304 10:6425 12:3388 6:0793 C 2 C76 ; C B B 10:9819 15:8931 12:4552 25:2244 23:1714 30:7154 C C B C B B 9:1120 9:5511 4:3438 3:6516 10:3746 40:1008 C A @ 5:1826 0

0:1172 B B 0:0502 B B B 0:1730 B B E ¼ B 0:4609 B B 0:4545 B B B 0:0635 @ 0:1754

3:7458

11:0221

19:1962

20:9471

0:3712 0:5810 0 0:0635 0

38:8931

1

C 0C C C 0:1092 1:2500 0 0:0342 0 C C C 0:7607 0:1909 0 0:0040 0 C 2 C76 ; C 0:3101 0:2946 0 0:0376 0 C C C 0:3513 0:7089 0 0:0339 0 C A

0:0947

0:6500

0

0:0457

0:2075

1:0256

0

0:0117

0

Table 1 The perturbation bound d depending on k1 and k2. k1k2 < 0 k1

k2

d

0.50000000000000 0.33333333333333 0.66666666666667 0.33333333333333 0.20000000000000 0.16666666666667 0.40000000000000 0.16666666666667 0.80000000000000 0.66666666666667 0.06250000000000 0.04000000000000 0.56250000000000 0.36000000000000 0.93750000000000 0.60000000000000

0.33333333333333 0.50000000000000 0.33333333333333 0.66666666666667 0.16666666666667 0.20000000000000 0.16666666666667 0.40000000000000 0.66666666666667 0.80000000000000 0.04000000000000 0.06250000000000 0.36000000000000 0.56250000000000 0.60000000000000 0.93750000000000

0.31483497699462 0.26564201183921 0.47225246549193 0.29515779093246 0.24596482577705 0.24287269653871 0.32795310103606 0.28335147929516 0.39354372124327 0.25501633136564 0.24179326233187 0.24123473297364 0.34541894618838 0.27128473431292 1.51120788957417 0.28593410996582

k1k2 > 0 k1 0.50000000000000 0.33333333333333 0.10000000000000 0.05000000000000 0.00100000000000 0.00100000000000 0.01000000000000 0.00100000000000 0.00010000000000 0.00001000000000 3.333333333333333e006 1.000000000000000e006 3.333333333333334e007

k2 0.50000000000000 0.33333333333333 0.10000000000000 0.10000000000000 0.10000000000000 0.02500000000000 0.01250000000000 0.02040816326531 0.02127659574468 0.02127659574468 0.02127659574468 0.02127659574468 0.02127659574468

d 0.70837869823789 0.47225246549193 0.28859872891173 0.27340932212691 0.25999885487544 0.24227166022484 0.24149273803565 0.24118632177684 0.24117431257607 0.24115260667088 0.24115099898148 0.24115043629525 0.24115027552824

ð2Þ

2 2 ? where d ¼ 1þk 1k1 kAT;S k2 kP S EP T kF .

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X. Liu et al. / Applied Mathematics and Computation 218 (2011) 4595–4604

where T ¼ C6 ; e ¼ ð0; 0;    ; 1Þ and S = span{e}. Then

0

ð2Þ

AT;S

0:0900

0:0283 0:0062

0:0029

0:0296

0:0619 0

B 0:1004 0:0449 0:0956 0:1114 0:0479 0:0463 B B B 0:0243 0:0202 0:0955 0:0663 0:0557 0:0821 ¼B B 0:0342 0:0729 0:0693 0:0043 0:0377 0:0167 B B @ 0:0581 0:1192 0:0553 0:0011 0:0128 0:0171 0:0058

0:0049 0:0237

0:0345

0:0241

0:0031

0:0103

0:0119

1

0C C C 0C C 0C C C 0A 0

and

0

ð2Þ

EAT;S

0:0089

0:0007

0:0242

B 0:0234 0:0243 0:0733 B B B 0:0237 0:0310 0:1268 B B ¼ B 0:0393 0:0506 0:0879 B B 0:0147 0:0253 0:0008 B B @ 0:0258 0:0037 0:0318 0:0622 0:0265 0:1195

0:0222 0

0:0539 0:0416

0:0601

0:0956 0:0804

0:1190

0:0987 0:0608

0:0795

0:0164 0:0114

0:0177

0:0080 0:0250 0:0907 0:0617

0:0465 0:0827

1

0C C C 0C C C 0 C: C 0C C C 0A 0

The following table shows the perturbation bounds d, obtained by Theorem 3.1, depending on k1 and k2, which satisfy the ð2Þ condition (3.1) with EAT;S . From Table 1, we find that the perturbation bound is better when either k1 < 0 and jk1j < k2 or k1, k2 > 0 and 1 kk21 (see the bold numbers in the Table 1). Acknowledgements The authors thank the first referee for helpful comments and suggestions. References [1] A. Ben-Israel, T.N.E. Generalized Inverses Theory and Applications, second ed., Springer Verlag, New York, 2003. [2] X. Chen, W. Li, A note on perturbation bounds of eigenspaces for Hermitian matrices, J. Comput. Appl. Math. 196 (2006) 338–346. [3] X. Chen, W. Li, Relative perturbation bounds for the subunitary polar factor under unitarily invariant norms, Adv. Math. 35 (2006) 178–184 (in Chinese). [4] Y. Chen, A Cramer rule for solution of the general restricted linear equation, Linear Multilinear Algebra 34 (1993) 177–186. [5] Y. Chen, X. Tan, Computing generalized inverses of matrices by iterative methods based on splittings of matrices, Appl. Math. Comput. 163 (2005) 309– 325. [6] J. Ding, New perturbation results on pseudo-inverse of linear operators in Banach spaces, Linear Algebra Appl. 362 (2003) 229–235. [7] L. Meng, B. Zheng, The optimal perturbation bounds of the Moore–Penrose inverse under Frobenius norm, Linear Algebra Appl. 423 (2010) 956–963. [8] G.W. Stewart, J. Sun, Matrix Perturbation Analysis, Academic Press, Boston, MA, 1990. [9] J. Sun, Matrix Perturbation Analysis, second ed., Science Press, Beijing, 2001 (in Chinese). ð2Þ [10] Y. Wei, H. Wu, On the perturbation and subproper splitting for the generalized inverse AT;S of rectangular matrixA, J. Comput. Appl. Math. 137 (2001) 317–329. ð2Þ [11] Y. Wei, G. Wang, Perturbation theory for the generalized inverse AT;S , J. Fudan Univ. (Nat. Sci.) 39 (2000) 482–488. ð2Þ [12] Y. Wei, N. Zhang, Condition number related with generalized inverse AT;S and constrained linear systems, J. Comput. Appl. Math. 157 (2003) 57–72. [13] X. Yang, Y. Wang, Some new perturbation theorems for generalized inverses of linear operators in Banach spaces, Linear Algebra Appl. 433 (2010) 1939–1949. ð2Þ [14] N. Zhang, Y. Wei, Perturbation bounds for the generalized inverse AT;S with application to constrained linear system, Appl. Math. Comput. 142 (2003) 63–78. [15] N. Zhang, Y. Wei, A note on the perturbation of an outer inverse, Calcolo 45 (2008) 263–273.