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The Representation and Approximation for Drazin Inverse

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The Representation and Approximation for Drazin Inverse View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 126 (2000) 417–432 www.elsevier.nl/locate/cam The representation and approximation for Drazin inverse ( Yimin Weia; ∗, Hebing Wub aDepartment of Mathematics and Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, People’s Republic of China bInstitute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China Received 2 August 1999; received in revised form 2 November 1999 Abstract We present a uniÿed representation theorem for Drazin inverse. Speciÿc expression and computational procedures for Drazin inverse can be uniformly derived. Numerical examples are tested and the application to the singular linear equations is also considered. c 2000 Elsevier Science B.V. All rights reserved. MSC: 15A09; 65F20 Keywords: Index; Drazin inverse; Representation; Approximation 1. Introduction The main theme of this paper can be described as a study of the Drazin inverse. In 1958, Drazin [4] introduced a di erent kind of generalized inverse in associative rings and semigroups that does not have the re exivity property but commutes with the element. Deÿnition 1.1. Let A be an n × n complex matrix, the Drazin inverse of A, denoted by AD,isthe unique matrix X satisfying Ak+1X = Ak ; XAX = X; AX = XA; (1.1) where k=Ind(A), the index of A, is the smallest nonnegative integer for which rank(Ak )=rank(Ak+1). ( Project 19901006 supported by National Nature Science Foundation of China, Doctoral Point Foundation of China and the Laboratory of Computational Physics in Beijing of China; the second author is also supported by the State Major Key Project for Basic Researches in China. ∗ Corresponding author. E-mail address: [email protected] (Y. Wei) 0377-0427/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S 0377-0427(99)00369-6 418 Y. Wei, H. Wu / Journal of Computational and Applied Mathematics 126 (2000) 417–432 Particularly, when Ind(A) = 1, the matrix X satisfying (1.1) is called the group inverse of A and is denoted by X = A#.IfA is nonsingular, then it is easily seen that Ind(A)=0 and A−1 satisÿes (1.1), i.e., AD = A−1. The Drazin inverse is very useful since various applications (for example, applications in singu- lar di erential and di erence equations, Markov chain, cryptography, iterative methods, multibody dynamics and numerical analysis) can be found in literature [1,2,5,8,12,14–23], respectively. In this paper, we shall present a general representation theorem for the Drazin inverse with a number of corollaries which result in speciÿc representation and computational procedures. These results are analogous to that of the Moore–Penrose inverse [7]. As usual, let R(A) be the range of A, N(A) be the null space of A. (A) denotes the spectrum of A and (A) denotes the spectral radius of A. The notation k·k stands for the spectral norm. 2. Representation for Drazin inverse In this section we establish a uniÿed representation theorem for the Drazin inverse, which may be viewed as an application of the classical theory summability to the representation of generalized inverse. The key to the representation theorem is the following lemma, due to Wei [14]. Lemma 2.1. Let A ∈ Cn×n with Ind(A)=k. Then; AD = A˜−1Al (l¿k); (2.1) ˜ l+1 l+1 k where A = A |R(Ak ) is the restriction of A on R(A ). Now, we are in a position to establish the representation theorem. n×n ˜ l+1 Theorem 2.2. Let A ∈ C with Ind(A)=k and denote A = A |R(Ak ) for l¿k. Suppose the ˜ spectrum of A is real. If is an open set such that (A) ⊂ ⊂(0; ∞) and {Sn(x)} is a family of ˜ continuous real valued functions on with limn→∞ Sn(x)=1=x uniformly on (A), then D ˜ l A = lim Sn(A)A : (2.2) n→∞ Furthermore; kS (A˜)Al − AD k n P 6 max |S (x)x − 1| +O();¿0; (2.3) D n kA kP x∈(A˜) −1 where P is an invertible matrix such that P AP is the Jordan canonical form of A and k A kP = kP−1AP k. Proof. Since (A) is real, we can always choose l (¿k) such that the spectrum of Al+1 is nonneg- ˜ l+1 ative and the spectrum of A = A |R(Ak ) is positive. Y. Wei, H. Wu / Journal of Computational and Applied Mathematics 126 (2000) 417–432 419 ˜ ˜−1 n×n By use of Theorem 10:27 in [13], we have limn→∞ Sn(A)=A uniformly in C . It follows from ˜ l ˜−1 l D l ˜ D Lemma 2.1 that limn→∞ Sn(A)A = A A = A . To obtain the error bound we note that A = AA and therefore ˜ l D ˜ ˜ D Sn(A)A − A =[Sn(A)A − I]A : Since P is nonsingular matrix such that P−1AP is the Jordan canonical form of A, it is a well-known fact that kAkP 6(A)+; thus kS (A˜)Al − AD k 6 kS (A˜)A˜ − I k kAD k n P " n P P # D 6 max |Sn(x)x − 1| +O() kA kP : x∈(A˜) In order to make use of this general error estimate above on speciÿc approximation procedures it will be convenient to have lower and upper bounds for (A˜). This is given in the next theorem. n×n k ˜ l+1 Theorem 2.3. Let A ∈ C with Ind(A)=k; rank(A )=r and denote A=A |R(Ak ) for l¿k. Suppose the spectrum of A is real. Then; for each ∈ (A˜); 1= kAD kl+1 66 kAkl+1 : (2.4) l+1 Proof. Let i (i =1; 2;:::;r) be the nonzero eigenvalues of A. Note that i is positive and 0 6= l+1 ˜ l+1 l+1 l+1 l+1 # D l+1 i ∈ (A) ⊆ (A ). It is obvious that Ind(A ) = 1 and 1=i ∈ [(A ) ]=[(A ) ]. We l+1 D l+1 arrive at 1=i 6 kA k , i.e., ¿1= kAD kl+1 for each ∈ (A˜). l+1 l+1 ˜ l+1 l+1 On the other hand, since k A k = k A |R(Ak ) k, one gets k A k 6 k A k , thus 6 k A k for each ∈ (A˜). The proof is complete. Remark 1. From the nonnegative matrix theory, we know that every real eigenvalue of the singular n×n n×n M-matrix A,(A ∈ Z = {A =(aij) ∈ R | aij60; ∀i; j =1; 2;:::;n; i 6= j}) is nonnegative. Remark 2. The numerical and symbolic computation of the Jordan canonical form of A can be found in the literature [3,6,9,10]. Remark 3. Since the index of A is not greater than n, we can always take l = n or n +1. 3. Approximation of Drazin inverse In this section, we present several corollaries which illustrate the use of Theorem 2.2 in develop- ing speciÿc representations and computational procedures for Drazin inverse and corresponding error bounds. 420 Y. Wei, H. Wu / Journal of Computational and Applied Mathematics 126 (2000) 417–432 P∞ A well-known summability method is called the Euler–Knopp method. A series n=0 an is said to be Euler–Knopp summable with parameter ¿0 to the value a if the sequence deÿned by Xn Xm m S = (1 − )m−j ja ; n j j m=0 j=0 converge to a.Ifa =(1− x)n for n =0; 1; 2;:::; then we obtain as the Euler–Knopp transform of P n ∞ n the series n=0 (1 − x) , the sequence given by Xn j Sn(x)= (1 − x) : j=0 Clearly limn→∞ Sn(x)=1=x uniformly on any compact subsets of the set E = {x: |1 − x| ¡ 1} = {x:0¡x¡2= }: Since, by Theorem 2.3 (A˜) ⊆ [1= kAD kl+1; kAkl+1 ] ⊆(0; kAkl+1 ]; −(l+1) l+1 if we choose the parameter ,0¡ ¡2 k A k such that (0; k A k ] ⊆ E , then we have the representation X∞ AD = (I − Al+1)nAl: (3.1) n=0 Note that if we set Xn l+1 j l An = (I − A ) A ; (3.2) j=0 then l l+1 l A0 = A ;An+1 =(I − A )An + A : (3.3) Therefore, we have the following corollary: Corollary 3.1. Let A ∈ Cn×n with Ind(A)=k and l¿k. Suppose the spectrum of A is real. Then D l+1 the sequence {An} deÿned by (3:3) converges to A ; if 0 ¡ ¡2= kAk . For the error bound we note that the sequence of functions {Sn(x)} satisÿes Sn+1(x)x − 1=(1− x)(Sn(x)x − 1): Thus, n+1 |Sn(x)x − 1| = |1 − x| : If x ∈ (A˜) and 0 ¡ ¡2= kAkl+1, then we see that |1 − x|6ÿ¡1; where ÿ = max{|1 − kAkl+1 |; |1 − kAD k−(l+1) |}. Therefore, n+1 |Sn(x)x − 1|6ÿ → 0(n →∞): (3.4) Y. Wei, H. Wu / Journal of Computational and Applied Mathematics 126 (2000) 417–432 421 It follows from (3.4) and Theorem 2.2 that the error bound is kA − AD k n P 6ÿn+1 +O(): (3.5) D kA kP To develop another iterative method, we regard 1=x as the root of the function s(y)=y−1 − x: The Newton–Raphson method can be used to approximate this root.
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