- Email: [email protected]
Condition numbers and structured perturbation of the W-weighted Drazin inverse
Applied Mathematics and Computation 165 (2005) 185–194 www.elsevier.com/locate/amc
Condition numbers and structured perturbation of the W-weighted Drazin inverse Tiangang Lei a, Yimin Wei a
b,*
, Ching-Wah Woo
c
Department of Mathematical and Physical Science, National Natural Science Foundation of China, Beijing 100085, PR China b Department of Mathematics, Fudan University, Shanghai 200433, PR China c Department of Mathematics, City University of Hong Kong, Hong Kong, PR China
Abstract We give the explicit condition number formulas for the W-weighted Drazin inverse of a rectangular matrix. We also present the structured perturbation of the W-weighted Drazin inverse, and consider the perturbation for the W-weighted Drazin inverse of a rectangular matrix and give an explicit expression for the W-weighted Drazin inverse of a perturbed matrix under some certain conditions. Finally, we discuss the minimum quality of the condition number. 2004 Elsevier Inc. All rights reserved. Keywords: Condition number; W-Weighted Drazin inverse; Perturbation
*
Corresponding author. E-mail addresses: [email protected] (T. Lei), [email protected] (Y. Wei), mawoo@ cityu.edu.hk (C.-W. Woo). 0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.044
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T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
1. Introduction It is well known that the classical normwise relative condition number measures the sensitivity of a matrix inverse. Let A 2 Cn · n be nonsingular matrix with a matrix norm k Æ k, this condition number can be defined as [5] condðAÞ ¼ lim
sup
!0þ kDAk 6 kAk
kðA þ DAÞ1 A1 k : kA1 k
ð1Þ
In order to reduce the sensitivity measure to a single number, two simplification have been introduced: 1. We look at the largest relative change in A1 compared with a relative change in A of size ; 2. We take the limit as ! 0+. Thus a condition number records the worst-case sensitivity to small perturbations. When the matrix norm is induced by a vector norm, it is well known that cond(A) has the characterization [6] condðAÞ ¼ jðAÞ ¼ kAk kA1 k:
ð2Þ
In this paper, we present a generalization of this fact to singular matrix A to obtain an analogous result for the W-weighted Drazin inverse Ad, W of A. The Drazin inverse is only defined for square matrices. Since the rectangular systems of differential equation arise, a W-weighted Drazin inverse for a rectangular matrix has been defined and studied by Cline and Greville [3]. If A and W are any complex matrices, m by n and n by m, respectively, then X = [(AW)D]2A is the unique solution to the following equations: ðAW Þ
kþ1
k
XW ¼ ðAW Þ ;
XWAWX ¼ X ;
AWX ¼ XWA;
ð3Þ
where k = Ind(AW), the index of AW, is the smallest nonnegative integer for which rank[(AW)k] = rank[(AW)k+1]. The matrix X is called the W-weighted Drazin inverse of A and is written as X = Ad, W [3]. In particular, when A becomes square matrix of order n, W reduces to the identity matrix In, the matrix X satisfying (3) is called the Drazin inverse [4] of A, and is denoted by X = AD. If A is nonsingular, then it is easily seen that Ind(A) = 0 and A1 satisfies (1), i.e., AD = A1. The Drazin inverse is very useful since various applications (for example, applications in singular differential and difference equations, Markov chains, iterative method and numerical analysis) were found in the literature [1,4,12]. As for the W-weighted Drazin inverse, we refer to [2,7–11,13]. In this paper, we give the explicit expression for the W-weighted Drazin inverse of a perturbed matrix under some certain conditions. We also consider
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
187
the structured perturbation of the W-weighted Drazin inverse, and give the formula of the condition number of the W-weighted Drazin inverse. Finally, we discuss the minimum quality of the condition number. As usual, let R(A) be the range of A, N(A) the null space of A. kAk2 and kAkF denotes the spectral and Frobenius norm of A, respectively. The following lemmas are very useful in Section 2. Lemma 1.1 [7,13]. Let B = A + E 2 Cm · n, W 2 Cn · m, k = max{Ind(AW), Ind(WA)}. If R(EW) R[(AW)k], R[(WE)*] R[(WA)k*], and kAd, W WEWk < 1. Then 1
ðA þ EÞd;W ¼ ½I þ Ad;W WEW Ad;W ; and kðA þ EÞd;W Ad;W k kAd;W WEW k 6 ; kAd;W k 1 kAd;W WEW k where k Æ k is any matrix norm that is induced by a vector norm. Lemma 1.2 [11]. Let A 2 Cm · n and W 2 Cn · m not be zero matrices. There exists two nonsingular matrices P 2 Cm · m, Q 2 Cn · n satisfying ! ! A11 0 W 0 11 A¼P Q1 ; W ¼ Q P 1 0 A22 0 W 22 where A11, W11 are both nonsingular, and A22W22, W22A22 are both strictly upper bi-diagonal matrix, and we have ! 1 ðW 11 A11 W 11 Þ 0 Ad;W ¼ P Q1 : r0 0 From Lemma 1.2, we give the following definition for fixed A, W and any matrix M, kMkP ;W ¼ kP 1 MWP k2 ; and ðF Þ
kMkP ;W ¼ kP 1 MWP kF : Þ It is obvious that kMkP, W and kMkðF P ;W satisfy the following property:
kM þ N kP ;W 6 kMkP ;W þ kN kP ;W ;
kaMkP ;W ¼ jaj kMkP ;W ;
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T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
and ðF Þ
ðF Þ
ðF Þ
kM þ N kP ;W 6 kMkP ;W þ kN kP ;W ;
ðF Þ
ðF Þ
kaMkP ;W ¼ jaj kMkP ;W :
2. Condition number We begin this section with a characterization of the k Æ kP, W––norm version of (1) and (2). Theorem 2.1. Let A, E 2 Cm · n, W 2 Cn · m, with k = max{Ind(AW), Ind(WA)}, the condition number condðAÞ ¼ lim
!0þ
sup kEkP ;W 6 kAkP ;W
kðA þ EÞd;W Ad;W kP ;W ; kAd;W kP ;W
ð4Þ
RðEW Þ R½ðAW Þk
R½ðWEÞ R½ðWAÞk
satisfies condðAÞ ¼ kAkP ;W kAd;W kP ;W :
ð5Þ
Proof. It follows from Lemma 1.1 and neglecting O(2) terms in a standard expansion gives ðA þ EÞd;W Ad;W ¼ Ad;W WEWAd;W : b using kEkP ;W 6 kAkP ;W ; we have k Ek b P ;W 6 1. Let E ¼ kAkP ;W E; First, we are establishing a basic inequality, 1 1 b b d;W k kAd;W W EWA Ad;W WPP 1 EWPP Ad;W WP k2 P ;W ¼ kP
b P ;W kAd;W kP ;W 6 kAd;W kP ;W k Ek ¼ kAd;W k2P ;W : Hence, the result is proved if we can show that 2 b d;W k sup ¼ kAd;W W EWA P ;W ¼ kAd;W kP ;W : kb E kP ;W 6 1
There exists vector y and x such that kðA11 W 11 Þ1 yk2 ¼ kðA11 W 11 Þ1 k2 ; where kyk2 = kxk2 = 1.
kx ðA11 W 11 Þ1 k2 ¼ kðA11 W 11 Þ1 k2 ;
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
Choosing " # y b ¼P E ½ x 0
" 0
W 1 11
0
0
0
189
# Q1 :
We can verify that 1 b b k Ek EWP k2 P ;W ¼ kP " # " y W 1 11
¼ ½x 0 0 0 " # y ¼ ½ x 0 0 2
0
#"
# W 22 0
W 11 0
0
2
¼ kyx k2 ¼ kyk2 kxk2 ¼ 1; and 1 1 b b d;w k kAd;w W EWA Ad;w WPP 1 EWPP Ad;w WP k2 P ;W ¼ kP " " # " 1 # # ðA W Þ1 0 y 0 W 11 11 11 ¼ ½ x 0 0 0 0 0 0 # " #" 1 W 11 0 0 ðA11 W 11 Þ 0 W 22 0 0 2 " #" # ðA W Þ1 y 0 x ðA W Þ1 0 11 11 11 11 ¼ 0 0 0 0 2 h ih i 1 1 ¼ ðA11 W 11 Þ y x ðA11 W 11 Þ 2
1
1
¼ kðA11 W 11 Þ yk2 kx ðA11 W 11 Þ k2 1
1
¼ kðA11 W 11 Þ k2 kðA11 W 11 Þ k2 2
2
¼ kP 1 Ad;W WP k2 ¼ kAd;W kP ;W : It is easy to check that
b ¼P EW
" # y 0
" ½ x
0
W 1 11
0
0
0
# Q1 Q
"
W 11
0
0
W 22
"
# P 1 ¼ P
yx
0
0
0
# P 1 ;
190
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
and
"
1
b ¼ Q W 11 yx W 11 WE 0
From
"
ðA11 W 11 Þk
0
0
0
ðW 11 A11 Þk
0
0
0
k
ðAW Þ ¼ P and
" k
ðWAÞ ¼ Q
# 0 1 Q : 0 # P 1 ;
# Q1 :
We can verify that b Þ R½ðAW Þ ; Rð EW k
and
b R½ðWAÞk : R½ðW EÞ We complete the proof.
h
Þ Next, we give a characterization of the k kðF P ;W norm version of (1) and (2).
Theorem 2.2. Let A,E 2 Cm · n,W 2 Cn · m, with k = max{Ind(AW), Ind(WA)}, the condition number condðF Þ ðAÞ ¼ lim
!0þ
Þ kðA þ EÞd;W Ad;W kðF P ;W
sup kEkP ;W 6 kAkP ;W RðEW Þ R½ðAW Þk
R½ðWEÞ R½ðWAÞk
ðF Þ
kAd;W kP ;W
;
ð6Þ
satisfies condðF Þ ðAÞ ¼
kAkPðF;WÞ kAd;W k2P ;W kAd;W kPðF;WÞ
:
ð7Þ
Proof. Using the same notation as in the proof of Theorem 2.1. The result is proved if we can show that b d;W kðF Þ ¼ kAd;W k2 : sup ¼ kAd;W W EWA P ;W P ;W kb E kP ;W 6 1
ð8Þ
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
191
The general inequalities kBCkF 6 kBk2kCkF and kBCkF 6 kBkFkCk2 give ‘‘ 6 ’’ in (8). Equality is found by taking " #
1 y 0 W 11 b ¼P E ½ x 0 Q1 : 0 0 0 By analogy of the proof of Theorem 2.1, we can show that this theorem.
h
3. Structured perturbation In this section, we will present a structured perturbation of the W-weighted Drazin inverse. This perturbation is close to component-wise perturbation. The notation jAj 6 jBj means that jaijj 6 jbijj for A = (aij) and B = (bij). Theorem 3.1. Let A 2 Cm · n, W 2 Cn · m, with k = max{Ind(AW), Ind(WA)}. If the perturbation has the special structure: jP1EWPj 6 jP1AWPj, jQ1WEQj 6 jQ1WAQj, and kAd,WkkWEWk < 1. Then 1
ðA þ EÞd;W ¼ ðI þ Ad;W WEW Þ Ad;W : Proof. Partition E ¼ P
E12 1 Q . E22
E11 E21
It follows from Lemma 1.2 and jP1EWPj 6 jP1AWPj that
E11 W 11 E12 W 22 A11 W 11 0 6 : E W 0 A22 W 22 E22 W 22 21 11 It is obvious that E21 W 11 ¼ 0;
jE22 W 22 j 6 jA22 W 22 j:
Since W11 is invertible and A22W22 is a strictly upper bi-diagonal matrix, we have E21 = 0, and E22W22 is a strictly upper bi-diagonal matrix. Similarly from jQ1WEQj 6 jQ1WAQj, we have E12 = 0 and W22E22 is also a strictly upper bi-diagonal matrix. After a little algebra, we get obtain the structure of A + E
A11 þ E11 0 AþE ¼P Q1 ; 0 A22 þ E22 and ðA þ EÞW ¼ P
ðA11 þ E11 ÞW 11
0
0
ðA22 þ E22 ÞW 22
P 1 :
192
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
From kAd,WWEWk2 6 kAd,Wk2kWEWk2 < 1, we obtain I + Ad,WWEW is nonsingular, i.e., " # 1 W 1 0 1 11 A11 ðA11 þ E11 ÞW 11 I þ Ad;W WEW ¼ P P 0 I is nonsingular. 1 Thus W 1 11 A11 ðA11 þ E11 ÞW 11 is nonsingular and A11 + E11 is also nonsingular, (A22 + E22)W22 is a strictly bi-diagonal matrix. Thus " # 1 1 0 1 W 1 D 2 11 ðA11 þ E 11 Þ W 11 ðA þ EÞd;W ¼ ð½ðA þ EÞW Þ ¼ P Q 0 0 1
¼ ðI þ Ad;W WEW Þ Ad;W :
4. The minimum quality of the condition number In this section, we will discuss the minimum quality of the condition number. Theorem 4.1. Let A 2 Cm · n, W 2 Cn · m with k = max{Ind(AW), Ind(WA)} and < kAd;W1k . If R(EW) R[(AW)k], R[(WE)* R[(WA)k*] and kEkP, W 6 . P ;W Suppose bðA;W ÞkEkP ;W
kðA þ EÞd;W Ad;W kP ;W kAkP ;W 6 : bðA;W ÞkEkP ;W kAd;W kP ;W 1 kAk
ð9Þ
P ;W
Then the condition number condðAÞ ¼ kAkP ;W kAd;W kP ;W 6 bðA; W Þ; where b(A, W) is a positive number independent of E, but dependent on A and W. This means that the condition number cond(A) is the minimum in the above sense.
Proof. It follows from the assumption and Lemma 1.1 that we have 1 X ðAd;W WEW Þi Ad;W ðA þ EÞd;W ¼ ðI þ Ad;W WEW Þ1 Ad;W ¼ i¼0
¼ Ad;W Ad;W WEWAd;W þ ðAd;W WEW Þ
2
1 X
i
ðAd;W WEW Þ Ad;W
i¼0
ð10Þ
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
193
So we obtain 3
2
kAd;W WEWAd;W kP ;W 6 kðA þ EÞd;W Ad;W kP ;W þ kAd;W kP ;W kEkP ;W 1 X i ðkAd;W kP ;W kEkP ;W Þ i¼0
Choose
y ½ x E ¼ P 0
" 0
W 1 11
0
0
0
# Q1 ;
where P, Q, W11, x and y are the same as Theorem 2.1.
1
y W 11 0 W 11 1
kEkP ;W ¼ kP EWP k2 ¼ ½x 0 0 0 0 0
yx 0
¼ 0 0 ¼ kyk2 kx k2 ¼ : 2
0 W 22 2
It is easy to verify that k
RðEW Þ R½ðAW Þ ; and kAd;W WEWAd;W kP ;W
k
R½ðWEÞ R½ðWAÞ ;
1
ðA11 W 11 Þ1 0 y W 11 0
¼ ½x 0 0 0 0 0 0
1 W 11 0 ðA11 W 11 Þ 0 0 W 22 0 0 2
1
ðA11 W 11 Þ y 0 x ðA11 W 11 Þ1 0 ¼ 0 0 0 0 2 h ih i 1 1
¼ ðA11 W 11 Þ y x ðA11 W 11 Þ 2 1 1 ¼ ðA11 W 11 Þ yk2 kx ðA11 W 11 Þ 1
2
1
¼ kðA11 W 11 Þ k2 kðA11 W 11 Þ k2 2 2 ¼ kP 1 Ad;W WP k2 ¼ kAd;W kP ;W : It follows form (9) and (10) that condðAÞ ¼ kAkP ;W kAd;W kP ;W ¼ kAkP ;W
kAd;W WEWAd;W kP ;W bðA; W Þ 6 bðA;W ÞkEk kEkP ;W kAd;W kP ;W 1 kAk P ;W P ;W
þ kAkP ;W kAd;W k2P ;W kEkP ;W
1 X ðkAd;W kP ;W kEkP ;W Þi : i¼0
Let kEkP, W = !0, we get cond(A) 6 b(A,W).
h
194
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
5. Concluding remarks We present the condition number of the W-weighted Drazin inverse of a rectangular matrix and also investigate the structured perturbation of it. It is natural to ask if we can extend these results to the Hilbert space or Banach space [7,8]. This will be the future topic for further research.
Acknowledgement Project supported by National Natural Science Foundation of China and Shanghai Education Committee.
References [1] S.L. Campbell, C.D. Meyer Jr., Generalized Inverses of Linear Transformation, Pitman, London, 1979, Dover, New York, 1991. [2] X. Chen, G. Chen, On the continuity and perturbation of W-weighted Drazin inverse, J. East China Normal Univ. 3 (1992) 20–26. [3] R.E. Cline, T.N.E. Greville, A Drazin inverse for rectangular matrices, Linear Algebra Appl. 29 (1980) 53–62. [4] M.P. Drazin, Pseudoinverses in associative rings and semigroups, Amer. Math. Monthly 65 (1958) 506–514. [5] G. Golub, C. Van Loan, Matrix Computations, third ed., Johns Hopkins University Press, Baltimore, 1996. [6] D. Higham, Condition numbers and their condition numbers, Linear Algebra Appl. 214 (1995) 193–213. [7] V. Rakoc˘evic´, Y. Wei, A weighted Drazin inverse and applications, Linear Algebra Appl. 350 (2002) 25–39. [8] V. Rakoc˘evic´, Y. Wei, The representation and approximation of the W-weighted Drazin inverse of linear operator in Hilbert space, Appl. Math. Comput. 141 (2003) 455–470. [9] G. Wang, Y. Wei, S. Qiao, Generalized Inverses: Theory and Computations, Science Press, Beijing/New York, 2004. [10] Y. Wei, A characterization for the W-weighted Drazin inverse and a Cramer rule for the W-weighted Drazin inverse solution, Appl. Math. Comput. 125 (2002) 303–310. [11] Y. Wei, Integral representation of the W-weighted Drazin inverse, Appl. Math. Comput. 144 (2003) 3–10. [12] Y. Wei, G. Wang, D. Wang, Condition number of Drazin inverse and their condition numbers of singular systems, Appl. Math. Comput. 146 (2003) 455–467. [13] Y. Wei, C.-W. Woo, T. Lei, A note on the perturbation of W-weighted Drazin inverse, Appl. Math. Comput. 149 (2004) 423–430.
Condition numbers and structured perturbation of the W-weighted Drazin inverse Tiangang Lei a, Yimin Wei a
b,*
, Ching-Wah Woo
c
Department of Mathematical and Physical Science, National Natural Science Foundation of China, Beijing 100085, PR China b Department of Mathematics, Fudan University, Shanghai 200433, PR China c Department of Mathematics, City University of Hong Kong, Hong Kong, PR China
Abstract We give the explicit condition number formulas for the W-weighted Drazin inverse of a rectangular matrix. We also present the structured perturbation of the W-weighted Drazin inverse, and consider the perturbation for the W-weighted Drazin inverse of a rectangular matrix and give an explicit expression for the W-weighted Drazin inverse of a perturbed matrix under some certain conditions. Finally, we discuss the minimum quality of the condition number. 2004 Elsevier Inc. All rights reserved. Keywords: Condition number; W-Weighted Drazin inverse; Perturbation
*
Corresponding author. E-mail addresses: [email protected] (T. Lei), [email protected] (Y. Wei), mawoo@ cityu.edu.hk (C.-W. Woo). 0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.044
186
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
1. Introduction It is well known that the classical normwise relative condition number measures the sensitivity of a matrix inverse. Let A 2 Cn · n be nonsingular matrix with a matrix norm k Æ k, this condition number can be defined as [5] condðAÞ ¼ lim
sup
!0þ kDAk 6 kAk
kðA þ DAÞ1 A1 k : kA1 k
ð1Þ
In order to reduce the sensitivity measure to a single number, two simplification have been introduced: 1. We look at the largest relative change in A1 compared with a relative change in A of size ; 2. We take the limit as ! 0+. Thus a condition number records the worst-case sensitivity to small perturbations. When the matrix norm is induced by a vector norm, it is well known that cond(A) has the characterization [6] condðAÞ ¼ jðAÞ ¼ kAk kA1 k:
ð2Þ
In this paper, we present a generalization of this fact to singular matrix A to obtain an analogous result for the W-weighted Drazin inverse Ad, W of A. The Drazin inverse is only defined for square matrices. Since the rectangular systems of differential equation arise, a W-weighted Drazin inverse for a rectangular matrix has been defined and studied by Cline and Greville [3]. If A and W are any complex matrices, m by n and n by m, respectively, then X = [(AW)D]2A is the unique solution to the following equations: ðAW Þ
kþ1
k
XW ¼ ðAW Þ ;
XWAWX ¼ X ;
AWX ¼ XWA;
ð3Þ
where k = Ind(AW), the index of AW, is the smallest nonnegative integer for which rank[(AW)k] = rank[(AW)k+1]. The matrix X is called the W-weighted Drazin inverse of A and is written as X = Ad, W [3]. In particular, when A becomes square matrix of order n, W reduces to the identity matrix In, the matrix X satisfying (3) is called the Drazin inverse [4] of A, and is denoted by X = AD. If A is nonsingular, then it is easily seen that Ind(A) = 0 and A1 satisfies (1), i.e., AD = A1. The Drazin inverse is very useful since various applications (for example, applications in singular differential and difference equations, Markov chains, iterative method and numerical analysis) were found in the literature [1,4,12]. As for the W-weighted Drazin inverse, we refer to [2,7–11,13]. In this paper, we give the explicit expression for the W-weighted Drazin inverse of a perturbed matrix under some certain conditions. We also consider
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
187
the structured perturbation of the W-weighted Drazin inverse, and give the formula of the condition number of the W-weighted Drazin inverse. Finally, we discuss the minimum quality of the condition number. As usual, let R(A) be the range of A, N(A) the null space of A. kAk2 and kAkF denotes the spectral and Frobenius norm of A, respectively. The following lemmas are very useful in Section 2. Lemma 1.1 [7,13]. Let B = A + E 2 Cm · n, W 2 Cn · m, k = max{Ind(AW), Ind(WA)}. If R(EW) R[(AW)k], R[(WE)*] R[(WA)k*], and kAd, W WEWk < 1. Then 1
ðA þ EÞd;W ¼ ½I þ Ad;W WEW Ad;W ; and kðA þ EÞd;W Ad;W k kAd;W WEW k 6 ; kAd;W k 1 kAd;W WEW k where k Æ k is any matrix norm that is induced by a vector norm. Lemma 1.2 [11]. Let A 2 Cm · n and W 2 Cn · m not be zero matrices. There exists two nonsingular matrices P 2 Cm · m, Q 2 Cn · n satisfying ! ! A11 0 W 0 11 A¼P Q1 ; W ¼ Q P 1 0 A22 0 W 22 where A11, W11 are both nonsingular, and A22W22, W22A22 are both strictly upper bi-diagonal matrix, and we have ! 1 ðW 11 A11 W 11 Þ 0 Ad;W ¼ P Q1 : r0 0 From Lemma 1.2, we give the following definition for fixed A, W and any matrix M, kMkP ;W ¼ kP 1 MWP k2 ; and ðF Þ
kMkP ;W ¼ kP 1 MWP kF : Þ It is obvious that kMkP, W and kMkðF P ;W satisfy the following property:
kM þ N kP ;W 6 kMkP ;W þ kN kP ;W ;
kaMkP ;W ¼ jaj kMkP ;W ;
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T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
and ðF Þ
ðF Þ
ðF Þ
kM þ N kP ;W 6 kMkP ;W þ kN kP ;W ;
ðF Þ
ðF Þ
kaMkP ;W ¼ jaj kMkP ;W :
2. Condition number We begin this section with a characterization of the k Æ kP, W––norm version of (1) and (2). Theorem 2.1. Let A, E 2 Cm · n, W 2 Cn · m, with k = max{Ind(AW), Ind(WA)}, the condition number condðAÞ ¼ lim
!0þ
sup kEkP ;W 6 kAkP ;W
kðA þ EÞd;W Ad;W kP ;W ; kAd;W kP ;W
ð4Þ
RðEW Þ R½ðAW Þk
R½ðWEÞ R½ðWAÞk
satisfies condðAÞ ¼ kAkP ;W kAd;W kP ;W :
ð5Þ
Proof. It follows from Lemma 1.1 and neglecting O(2) terms in a standard expansion gives ðA þ EÞd;W Ad;W ¼ Ad;W WEWAd;W : b using kEkP ;W 6 kAkP ;W ; we have k Ek b P ;W 6 1. Let E ¼ kAkP ;W E; First, we are establishing a basic inequality, 1 1 b b d;W k kAd;W W EWA Ad;W WPP 1 EWPP Ad;W WP k2 P ;W ¼ kP
b P ;W kAd;W kP ;W 6 kAd;W kP ;W k Ek ¼ kAd;W k2P ;W : Hence, the result is proved if we can show that 2 b d;W k sup ¼ kAd;W W EWA P ;W ¼ kAd;W kP ;W : kb E kP ;W 6 1
There exists vector y and x such that kðA11 W 11 Þ1 yk2 ¼ kðA11 W 11 Þ1 k2 ; where kyk2 = kxk2 = 1.
kx ðA11 W 11 Þ1 k2 ¼ kðA11 W 11 Þ1 k2 ;
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
Choosing " # y b ¼P E ½ x 0
" 0
W 1 11
0
0
0
189
# Q1 :
We can verify that 1 b b k Ek EWP k2 P ;W ¼ kP " # " y W 1 11
¼ ½x 0 0 0 " # y ¼ ½ x 0 0 2
0
#"
# W 22 0
W 11 0
0
2
¼ kyx k2 ¼ kyk2 kxk2 ¼ 1; and 1 1 b b d;w k kAd;w W EWA Ad;w WPP 1 EWPP Ad;w WP k2 P ;W ¼ kP " " # " 1 # # ðA W Þ1 0 y 0 W 11 11 11 ¼ ½ x 0 0 0 0 0 0 # " #" 1 W 11 0 0 ðA11 W 11 Þ 0 W 22 0 0 2 " #" # ðA W Þ1 y 0 x ðA W Þ1 0 11 11 11 11 ¼ 0 0 0 0 2 h ih i 1 1 ¼ ðA11 W 11 Þ y x ðA11 W 11 Þ 2
1
1
¼ kðA11 W 11 Þ yk2 kx ðA11 W 11 Þ k2 1
1
¼ kðA11 W 11 Þ k2 kðA11 W 11 Þ k2 2
2
¼ kP 1 Ad;W WP k2 ¼ kAd;W kP ;W : It is easy to check that
b ¼P EW
" # y 0
" ½ x
0
W 1 11
0
0
0
# Q1 Q
"
W 11
0
0
W 22
"
# P 1 ¼ P
yx
0
0
0
# P 1 ;
190
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
and
"
1
b ¼ Q W 11 yx W 11 WE 0
From
"
ðA11 W 11 Þk
0
0
0
ðW 11 A11 Þk
0
0
0
k
ðAW Þ ¼ P and
" k
ðWAÞ ¼ Q
# 0 1 Q : 0 # P 1 ;
# Q1 :
We can verify that b Þ R½ðAW Þ ; Rð EW k
and
b R½ðWAÞk : R½ðW EÞ We complete the proof.
h
Þ Next, we give a characterization of the k kðF P ;W norm version of (1) and (2).
Theorem 2.2. Let A,E 2 Cm · n,W 2 Cn · m, with k = max{Ind(AW), Ind(WA)}, the condition number condðF Þ ðAÞ ¼ lim
!0þ
Þ kðA þ EÞd;W Ad;W kðF P ;W
sup kEkP ;W 6 kAkP ;W RðEW Þ R½ðAW Þk
R½ðWEÞ R½ðWAÞk
ðF Þ
kAd;W kP ;W
;
ð6Þ
satisfies condðF Þ ðAÞ ¼
kAkPðF;WÞ kAd;W k2P ;W kAd;W kPðF;WÞ
:
ð7Þ
Proof. Using the same notation as in the proof of Theorem 2.1. The result is proved if we can show that b d;W kðF Þ ¼ kAd;W k2 : sup ¼ kAd;W W EWA P ;W P ;W kb E kP ;W 6 1
ð8Þ
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
191
The general inequalities kBCkF 6 kBk2kCkF and kBCkF 6 kBkFkCk2 give ‘‘ 6 ’’ in (8). Equality is found by taking " #
1 y 0 W 11 b ¼P E ½ x 0 Q1 : 0 0 0 By analogy of the proof of Theorem 2.1, we can show that this theorem.
h
3. Structured perturbation In this section, we will present a structured perturbation of the W-weighted Drazin inverse. This perturbation is close to component-wise perturbation. The notation jAj 6 jBj means that jaijj 6 jbijj for A = (aij) and B = (bij). Theorem 3.1. Let A 2 Cm · n, W 2 Cn · m, with k = max{Ind(AW), Ind(WA)}. If the perturbation has the special structure: jP1EWPj 6 jP1AWPj, jQ1WEQj 6 jQ1WAQj, and kAd,WkkWEWk < 1. Then 1
ðA þ EÞd;W ¼ ðI þ Ad;W WEW Þ Ad;W : Proof. Partition E ¼ P
E12 1 Q . E22
E11 E21
It follows from Lemma 1.2 and jP1EWPj 6 jP1AWPj that
E11 W 11 E12 W 22 A11 W 11 0 6 : E W 0 A22 W 22 E22 W 22 21 11 It is obvious that E21 W 11 ¼ 0;
jE22 W 22 j 6 jA22 W 22 j:
Since W11 is invertible and A22W22 is a strictly upper bi-diagonal matrix, we have E21 = 0, and E22W22 is a strictly upper bi-diagonal matrix. Similarly from jQ1WEQj 6 jQ1WAQj, we have E12 = 0 and W22E22 is also a strictly upper bi-diagonal matrix. After a little algebra, we get obtain the structure of A + E
A11 þ E11 0 AþE ¼P Q1 ; 0 A22 þ E22 and ðA þ EÞW ¼ P
ðA11 þ E11 ÞW 11
0
0
ðA22 þ E22 ÞW 22
P 1 :
192
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From kAd,WWEWk2 6 kAd,Wk2kWEWk2 < 1, we obtain I + Ad,WWEW is nonsingular, i.e., " # 1 W 1 0 1 11 A11 ðA11 þ E11 ÞW 11 I þ Ad;W WEW ¼ P P 0 I is nonsingular. 1 Thus W 1 11 A11 ðA11 þ E11 ÞW 11 is nonsingular and A11 + E11 is also nonsingular, (A22 + E22)W22 is a strictly bi-diagonal matrix. Thus " # 1 1 0 1 W 1 D 2 11 ðA11 þ E 11 Þ W 11 ðA þ EÞd;W ¼ ð½ðA þ EÞW Þ ¼ P Q 0 0 1
¼ ðI þ Ad;W WEW Þ Ad;W :
4. The minimum quality of the condition number In this section, we will discuss the minimum quality of the condition number. Theorem 4.1. Let A 2 Cm · n, W 2 Cn · m with k = max{Ind(AW), Ind(WA)} and < kAd;W1k . If R(EW) R[(AW)k], R[(WE)* R[(WA)k*] and kEkP, W 6 . P ;W Suppose bðA;W ÞkEkP ;W
kðA þ EÞd;W Ad;W kP ;W kAkP ;W 6 : bðA;W ÞkEkP ;W kAd;W kP ;W 1 kAk
ð9Þ
P ;W
Then the condition number condðAÞ ¼ kAkP ;W kAd;W kP ;W 6 bðA; W Þ; where b(A, W) is a positive number independent of E, but dependent on A and W. This means that the condition number cond(A) is the minimum in the above sense.
Proof. It follows from the assumption and Lemma 1.1 that we have 1 X ðAd;W WEW Þi Ad;W ðA þ EÞd;W ¼ ðI þ Ad;W WEW Þ1 Ad;W ¼ i¼0
¼ Ad;W Ad;W WEWAd;W þ ðAd;W WEW Þ
2
1 X
i
ðAd;W WEW Þ Ad;W
i¼0
ð10Þ
T. Lei et al. / Appl. Math. Comput. 165 (2005) 185–194
193
So we obtain 3
2
kAd;W WEWAd;W kP ;W 6 kðA þ EÞd;W Ad;W kP ;W þ kAd;W kP ;W kEkP ;W 1 X i ðkAd;W kP ;W kEkP ;W Þ i¼0
Choose
y ½ x E ¼ P 0
" 0
W 1 11
0
0
0
# Q1 ;
where P, Q, W11, x and y are the same as Theorem 2.1.
1
y W 11 0 W 11 1
kEkP ;W ¼ kP EWP k2 ¼ ½x 0 0 0 0 0
yx 0
¼ 0 0 ¼ kyk2 kx k2 ¼ : 2
0 W 22 2
It is easy to verify that k
RðEW Þ R½ðAW Þ ; and kAd;W WEWAd;W kP ;W
k
R½ðWEÞ R½ðWAÞ ;
1
ðA11 W 11 Þ1 0 y W 11 0
¼ ½x 0 0 0 0 0 0
1 W 11 0 ðA11 W 11 Þ 0 0 W 22 0 0 2
1
ðA11 W 11 Þ y 0 x ðA11 W 11 Þ1 0 ¼ 0 0 0 0 2 h ih i 1 1
¼ ðA11 W 11 Þ y x ðA11 W 11 Þ 2 1 1 ¼ ðA11 W 11 Þ yk2 kx ðA11 W 11 Þ 1
2
1
¼ kðA11 W 11 Þ k2 kðA11 W 11 Þ k2 2 2 ¼ kP 1 Ad;W WP k2 ¼ kAd;W kP ;W : It follows form (9) and (10) that condðAÞ ¼ kAkP ;W kAd;W kP ;W ¼ kAkP ;W
kAd;W WEWAd;W kP ;W bðA; W Þ 6 bðA;W ÞkEk kEkP ;W kAd;W kP ;W 1 kAk P ;W P ;W
þ kAkP ;W kAd;W k2P ;W kEkP ;W
1 X ðkAd;W kP ;W kEkP ;W Þi : i¼0
Let kEkP, W = !0, we get cond(A) 6 b(A,W).
h
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5. Concluding remarks We present the condition number of the W-weighted Drazin inverse of a rectangular matrix and also investigate the structured perturbation of it. It is natural to ask if we can extend these results to the Hilbert space or Banach space [7,8]. This will be the future topic for further research.
Acknowledgement Project supported by National Natural Science Foundation of China and Shanghai Education Committee.
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