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The Representation and Computation of Generalized Inverse AT,S

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The Representation and Computation of Generalized Inverse AT,S CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 213 (2008) 248–257 www.elsevier.com/locate/cam (2) ଁ The representation and computation of generalized inverse AT,S Xingping Shenga,b,∗, Guoliang Chena, Yi Gonga,c aDepartment of Mathematics, East China Normal University, Shanghai 200062, China bDepartment of Mathematics, Fuyang Normal College, Fuyang, Anhui 236032, China cFundamental Department of Shanghai Customs College, Shanghai 200002, China Received 30 June 2006; received in revised form 5 January 2007 Abstract (2) This paper presents a novel representation for the generalized inverse AT,S. Based on this, we give an algorithm to compute this A† generalized inverse. As an application, we use Gauss–Jordan elimination to compute the weighted Moore–Penrose inverse M,N and the Drazin inverse Ad. © 2007 Elsevier B.V. All rights reserved. MSC: 15A09; 65F10 Keywords: Gauss–Jordan elimination; Generalized inverse; Row operation 1. Introduction It is well-known fact that the common six important generalized inverses: the Moore–Penrose inverse A†, the weight A† A A A(−1) Moore–Penrose inverse M,N, the Drazin inverse d, the group inverse g, the Bott–Duffin inverse L and the (†) (2) generalized Bott–Duffin inverse AL are all generalized inverse AT,S, which having the prescribed range T and the null space S of {2}—(or outer) inverse of A. The {2} inverse has many applications, for example, the application in the iterative methods for solving the nonlinear equations [2,8] and the application to statistics [4–6]. In particular, {2} inverse plays an important role in stable approximation of ill-posed problems and in linear and nonlinear problems involving rank-deficient generalized inverse [7,9] It is celebrated that Gauss–Jordan elimination procedure compute the inverse of a nonsingular matrix A by executing elementary row operations on the pair (A, I) to transform it into (I, A−1). Moreover Gauss–Jordan elimination can be used to determine whether or not a matrix is nonsingular. However, one cannot directly use this method on a generalized ଁ This project was granted financial support from China Postdoctoral Science Foundation (No. 20060400634), Shanghai Science and Technology Committee (No. 062112065) and The University Young Teacher Sciences Foundation of Anhui Province (No. 2006jql220zd). ∗ Corresponding author. Department of Mathematics, East China Normal University, Shanghai 200062, China. E-mail address: [email protected] (X. Sheng). 0377-0427/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2007.01.009 X. Sheng et al. / Journal of Computational and Applied Mathematics 213 (2008) 248–257 249 inverse of a rectangular matrix or a square singular matrix A. In 1987 Anstreicher [1] used this way to compute the index, generalized null spaces, and Drazin inverse. In this paper, we first give explicit expression for the group inverse (2) Ag. Based on this, we establish a novel representation for AT,S. As an application, we use Gauss–Jordan elimination to compute the common important six kinds generalized inverse. 2. Notation and preliminaries n m×n Throughout this paper the following notation are used: C denotes the n dimensional complex space. Cr denotes the space of m × n complex matrices with rank r.ForA ∈ Cm×n, R(A) and N(A) denote the range and null space of A. r(A) and Ind(A) denote the rank and the index of A. A∗ and A# = N −1A∗M denote the conjugate transpose P A†,A† ,A ,A ,A(−1),A(†) and weighted conjugate transpose of A. L denotes the orthogonal projection on L. M,N d g L L denote the M–P inverse, weighted M–P inverse, Drazin inverse, group inverse, Bott–Duffin inverse and generalized Bott–Duffin inverse, respectively. (2) In 1974, Ben-Israel [2] gave a definition of AT,S, which as follows: Lemma 2.1 (Ben-Israel [2]). Let A ∈ Cm×n be of rank r, let T be a subspace of Cn of dimension s r, and let S be a subspace of Cm of dimension m − s. Then A has a {2} inverse X such that R(X) = T and N(X)= S, if and only if m AT ⊕ S = C . (2) In which case X is unique. This X is denoted by AT,S. A†,A† ,A ,A ,A(−1) A(†) The next lemma shows that the common six kinds of generalized inverse: M,N d g L and L are all (2) generalized AT,S (for which exists a matrix G such that R(G) = T and N(G) = T ). A ∈ Cm×n A† A† Lemma 2.2. (1) Let . Then for the Moore–Penrose inverse and the weight Moore–Penrose inverse M,N, one has † (2) (a) [2]A = AR(A∗),N(A∗), A† = A(2) (b) [2] M,N R(A#),N(A#), where M and N are Hermitian positive definite matrices of order m and n, respectively. In addition A# = N −1AM. n×n (−1) (2) Let A ∈ C . Then for the Drazin inverse Ad, the group inverse Ag, the Bott–Duffin inverse AL and the (†) generalized Bott–Duffin inverse AL , one has (2) (c) [2]Ag = AR(A),N(A), if Ind(A) = 1, A = A(2) (A) = k (d) [2] d R(Ak),N(Ak), if Ind , A(−1) = A(2) Cn AL ⊕ L⊥ = Cn (e) [2] L L,L⊥ , where L is a subspace of and satisfies , A(†) = A(2) Cn S = R(P A) L − . (f) [3] L S,S⊥ , where L is a subspace of L , and A is p s d matrix. (2) In 1998, Wei [10] gave an expression of the generalized AT,S by using group inverse, this gave a new method to (2) study AT,S. Lemma 2.3 (Wei [10]). Let A ∈ Cm×n be of rank r, let T be a subspace of Cn of dimension s r, and let S be a subspace of Cm of dimension m − s. In addition, suppose G ∈ Cn×m such that R(G) = T and N(G) = S. If A has a (2) {2} inverse of AT,S then Ind(AG) = Ind(GA) = 1. 250 X. Sheng et al. / Journal of Computational and Applied Mathematics 213 (2008) 248–257 Further we have (2) AT,S = G(AG)g = (GA)gG. The next Lemma will be used repeatedly in the following section. Lemma 2.4. Let A ∈ Cn×n with rank = r and Ind(A) = 1. Then E1 r×n (n−r)×n (1) there exists an elementary row operation matrix E = , where E ∈ Cr and E ∈ C , such that E2 1 2 (n−r) R(A) = N(E ), R(A∗E∗) = R(A∗) 2 1 , (2.1) −1 E1A E1 A2 = A. (2.2) E2 0 n×r n×(n−r) (2) There also exists an elementary column operation matrix F = (F1 F2), where F1 ∈ Cr and F2 ∈ C(n−r) , such that R(A∗) = N(F∗), R(AF ) = R(A) 2 1 , (2.3) 2 −1 A (F1 0)(AF 1 F2) = A. (2.4) Proof. (1) Under the condition r(A) = r, we execute elementary row operations on the matrix A to transform it into E Hermitian normal form. That is, there exists an elementary row matrix E = 1 such that E2 E1A E1A EA = = , E2A 0 r×n (n−r)×n r×n where E1 ∈ Cr , E2 ∈ C(n−r) and E1A ∈ Cr . From which we can get that r(A) = r(E1A) and E2A = 0. Then R(A∗) = R(A∗E∗) 1 (2.5) and R(A) ⊂ N(E2). According to the fact r(A) = r = n − r(E2),wehave R(A) = N(E2). E A In the next part, we first show the matrix 1 is nonsingular. E2 By the condition Ind(A) = 1, then R(A) ⊕ N(A) = Cn. Further we have ∗ ∗ n R(A ) ⊕ N(A ) = C . Then it follows from (2.5) that Cn = R(A∗) ⊕ N(A∗) = R(A∗E∗) ⊕ R(A)⊥ = R(A∗E∗) ⊕ N(E )⊥ = R(A∗E∗) ⊕ R(E∗) 1 1 2 1 2 . E A Hence the matrix 1 is invertible. E2 X. Sheng et al. / Journal of Computational and Applied Mathematics 213 (2008) 248–257 251 On the other hand, we have 2 2 E1A E1A E1A E1 A = = = A2. E2 E2A 0 0 That is −1 E1A E1 A2 = A. E2 0 If we use this way to A∗, Eqs. (2.3) and (2.4) hold. This completes the proof. 3. Main results (2) In this section, we will first compute the group inverse, then give an expression of the generalized inverse AR(G),N(G), which having the prescribed range R(G) and the null space N(G), based on Lemma 2.3. Theorem 3.1. Let A ∈ Cn×n be of rank r and Ind(A) = 1, then E (1) There exists an elementary row matrix E = 1 , such that E2 −1 2 E1A E1 Ag = A. (3.1) E2 0 (2) There also exists an elementary column matrix F = (F1 F2), such that −1 2 Ag = A((F1 0)(AF 1 F2) ) , (3.2) where E1,E2,E1A and F1, F2, AF 1 are the same as in Lemma 2.4. − 2 E A 1 E Proof. (1) We denote A= 1 1 A. E2 0 By the condition Ind(A) = 1, we have n R(A) ⊕ N(A) = C , (3.3) R(A) = R(A2) (3.4) from (3.3), for any x ∈ Cn, write x = PR(A),N(A)x + PN(A),R(A)x = x1 + x2, where x1 ∈ R(A), x2 ∈ N(A).
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