Drazin Inverse

A new result on the Drazin inverse of 2 × 2 block matrix M = A B D C , where A and C are square matrices are presented, extended in the case when D = 0, the well known representation for the Drazin inverse of M , given by Hartwig, Meyer and Rose in 1977, respectively. Using that new result, an explicit representation for the Drazin inverse of a modified matrix P + RS and its generalized matrix P Q + RS are also presented.

Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if A d denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B) d is computed in terms of A d and B d . Thus, recent results of R. E. Hartwig, G. Wang and Y. Wei are extended to infinite dimensional settings with simplified proof. 2000 Mathematics subject classification: 47A05, 47A55, 15A09.

We generalize the iterative method for implementation of the limit representation of the Moore±Penrose inverse, lim a30 as e à e À 1 e à , introduced by Zukovski and Lipcer [ Z. Vicisl. Mat. i Mat. Fiz. 12 (1972) 843; 15 (1975) 489]. More precisely, we de®ne an iterative process for implementing the general limit formula lim a30 as à À1 à . In a certain case the presented method gives an iterative process which implements the known limit representation of the Drazin inverse, introduced by Meyer [SIAM J. Appl. Math. 26 (1974) 469]. Also, the known limit representation of the generalized inverse e 2 Y is a special case of the general limit formula. In the case of a full rank matrix e, we de®ne a limit representation and an iterative method for computation of the left or right inverses of e. Also, we introduce two alternative limit representations for the set of {2} and {1,2}-inverses, and develop the corresponding iterative processes. Moreover, we propose a ®nite algorithm for computation of the generalized inverses contained in the limit formula lim a30 as à À1 à , using the method developed by Ji [Appl. Math. Comput. 61 (1994) 151]. Ó Leverrier±Faddeev algorithm 0096-3003/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 -3 0 0 3 ( 9 8 ) 1 0 0 4 8 -6

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

over integral domains. The main results consist of necessary and sufficient conditions for the existence of a group inverse, a new formula for a group inverse when it exists, and necessary and sufficient cpnditions for the existence of a Drazin inverse. We show that a square matrix A of rank r over an integral domain W has a group inverse if and only if the sum of all r x r principal minors of A is an invertible element of R. We also show that the group inverse of A when it exists is a polynomial in A with coefficients from W. 'J

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Some additive perturbation results for Drazin inverses are given. In particular, a formula is given for the Drazin inverse of a sum of two matrices, when one of the products of these matrices vanishes. Some special applications of this are also considered.

Properties of the Drazin inverse of the matrix F ¼ I I E 0 , with E square, are investigated. Based on this approach, it is obtained an explicit formula for the Drazin inverse of matrices of the form , where U, V, P and Q are n • k. The representation for A D is given in terms of k • k matrices involving the individual blocks under some conditions. Some special cases are also analyzed.

In this talk results on isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be presented. The case of Calkin algebras on Banach and Hilbert spaces will be also studied. In addition, (generalized) B-Fredholm elements in Banach algebras will be considered.

Given Banach spaces $X$ and $Y$ and Banach space operators $A\in L(X)$ and $B\in L(Y)$, let $\rho\colon L(Y,X)\to L(Y,X)$ denote  the generalized derivation defined by $A$ and $B$, i.e., $\rho (U)=AU-UB$ ($U\in L(Y,X)$). The main objective of this article is to study Weyl and  Browder type theorems for $\rho\in L(L(Y,X))$. To this end, however, first the isolated points of the spectrum and the Drazin spectrum of $\rho\in L(L(Y,X))$ need to be characterized. In addition, it will be also proved that if $A$ and $B$ are polaroid (respectively isoloid), then $\rho$ is polaroid (respectively isoloid).

In this article, we introduce a full-rank representation of the Drazin inverse A D of a given complex matrix A, which is based on an arbitrary full-rank decomposition of A l , l k, where k is the index of A. Using this general representation, we introduce a determinantal representation of the Drazin inverse. More precisely, we represent elements of the Drazin inverse A D as a fraction of two expressions involving minors of the order rank(A k ), k = ind(A), taken from the matrices A and rank invariant powers A l , l k. Also, we examine conditions for the existence of the Drazin inverse for matrices whose elements are taken from an integral domain. Finally, a few correlations between the minors of the Drazin inverse A D , powers of the Drazin inverse and the minors of the matrix A k , k = ind(A), are explicitly derived. (P.S. Stanimirović). 0024-3795/00/$ -see front matter 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 -3 7 9 5 ( 0 0 ) 0 0 0 7 5 -6 N k = Q k,m × Q k,n , where k rank(A);

Given a Banach Algebra $A$ and $a\in A$, several relationships among the Drazin spectrum of $a$ and the ascent, the descent and the Drazin spectra of the multiplication operators $L_a$ and $R_a$ will be presented; the Banach space operator case will be also examined. In addition, a characterization of the spectrum of $a$ in terms of the Drazin spectrum and the poles of the resolvent of $a$ will be considered. Furthermore, several basic properties of the Drazin spectrum in Banach algebras will be studied.

Two representations for the Drazin inverse of a 2×2 block matrix M = [ A C B D ], where A and D are square matrices, in terms of the Drazin inverses of A and D have been recently developed under the assumptions that C(I − AA D ) = 0 and (I − AA D )B = 0, and that the generalized Schur complement D −CA D B is either nonsingular or zero. These two representations of M D are extended to the case where C(I − AA D ) = 0 and (I − AA D )B = 0 are substituted with C(I − AA D )B = 0 and A(I − AA D )B = 0. Moreover, upper bounds for the index of M are studied. Numerical examples are given to illustrate the new results.

In this paper, we investigate additive results of the Drazin inverse of elements in a ring R. Under the condition ab = ba, we show that a + b is Drazin invertible if and only if aa D (a + b) is Drazin invertible, where the superscript D means the Drazin inverse. Furthermore we find an expression of (a + b) D . As an application we give some new representations for the Drazin inverse of a 2 × 2 block matrix.

Given a Banach Algebra A and $a\in A$, several relations among the Drazin spectrum of $a$ and the Drazin spectra of the multiplication operators $L_a$ and $R_a$ will be stated. The Banach space operator case will be also examined. Furthermore, a characterization of the Drazin spectrum will be considered.

In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.

We study perturbations of the Drazin inverse of a closed linear operator A for the case when the perturbed operator has the same spectral projection as A. This theory subsumes results recently obtained by Wei and Wang, Rakočević and Wei, and Castro and Koliha. We give explicit error estimates for the perturbation of Drazin inverse, and error estimates involving higher powers of the operators.

In the literature, an important class of generalized inverse matrices corresponds to the group inverse, that is, matrices of index 1. Recently, the nonnegativity of a singular system has been applied to different fields. In this paper, an algorithm to check the nonnegativity of a singular linear control system of index 1 is presented. To this purpose, the nonnegativity of this kind of systems is characterized using a block partition of the original matrices. In this way, we can work with matrices having smaller sizes and keeping the original information. Finally, numerical examples illustrating the obtained results are shown.

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [A. Schuster, P. Hippe, Inversion of polynomial matrices by interpolation, IEEE Trans. Automat. Control 37 (3) (1992) 363-365], applicable to usual matrix inverse. Also, we improve our interpolation method, using a more effective estimation of degrees of polynomial matrices generated in Leverrier-Faddeev method. Algorithms are implemented and tested in the symbolic package MATHEMATICA.

We investigate two algorithms for computing the Moore-Penrose and Drazin inverse of a given one-variable polynomial matrix by interpolation. These algorithms differ in the method used for constant matrices inverses computing. The first algorithm uses the Grevile's method, and the second one uses the Leverrier-Faddeev method and its generalization. These algorithms are especially useful for symbolic computation in procedural programming languages. We compare results by implementing the algorithms in the programming package MATHEMATICA and in the procedural programming languages DELPHI and C++.

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [A. Schuster, P. Hippe, Inversion of polynomial matrices by interpolation, IEEE Trans. Automat. Control 37 (3) (1992) 363–365], applicable to usual matrix inverse. Also, we improve our interpolation method, using a more

over integral domains. The main results consist of necessary and sufficient conditions for the existence of a group inverse, a new formula for a group inverse when it exists, and necessary and sufficient cpnditions for the existence of a Drazin inverse. We show that a square matrix A of rank r over an integral domain W has a group inverse if and only if the sum of all r x r principal minors of A is an invertible element of R. We also show that the group inverse of A when it exists is a polynomial in A with coefficients from W. 'J

Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore–Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

Starting from algorithms introduced in [Ky M. Vu, An extension of the Faddeev's algorithms, in: Proceedings of the IEEE Multi-conference on Systems and Control on September 3-5th, 2008, San Antonio, TX] which are applicable to one-variable regular polynomial matrices, we introduce two dual extensions of the Faddeev's algorithm to one-variable rectangular or singular matrices. Corresponding algorithms for symbolic computing the Drazin and the Moore-Penrose inverse are introduced. These algorithms are alternative with respect to previous representations of the Moore-Penrose and the Drazin inverse of one-variable polynomial matrices based on the Leverrier-Faddeev's algorithm. Complexity analysis is performed. Algorithms are implemented in the symbolic computational package MATHEMATICA and illustrative test examples are presented.

Three representations for the W-weighted Drazin inverse of a modified matrix A À CB have been developed under some conditions where A 2 C mÂn , W 2 C nÂm , B 2 C pÂn , and C 2 C mÂp. The results of the paper not only extend the earlier similar works about the Drazin inverse and group inverse, but also weaken the assumed condition of a result of the Drazin inverse to the case where C d ZZ g ¼ Z g ZC d is substituted with CðC d ZZ g À Z g ZC d ÞB ¼ 0. The perturbation bound for the Drazin inverse is also established as a direct corollary of the representations. Numerical examples are given to illustrate the new results.

Let A π denote the eigenprojection of a matrix A corresponding to the eigenvalue 0. We characterize matrices B such that B π = A π , and derive from the results: (1) error bounds for the Drazin inverse of a perturbation, (2) improvement of error bounds given recently by Wei [10], Wei and Wang [11] and Castro González et al. [4], and (3) a new characterization of EP matrices.

We present a unified representation theorem for the class of all outer generalized inverses of a bounded linear operator. Using this representation we develop a few specific expressions and computational procedures for the set of outer generalized inverses. The obtained result is a generalization of the well-known representation theorem of the Moore--Penrose inverse as well as a generalization of the

If A(z) is a function of a complex variable with values in the space B(X) of all bounded linear operators on a Banach space X with each A(z) g-Drazin invertible, we study the conditions under which the g-Drazin inverse A D (z) is holomorphic and finite-meromorphic. From our results we recover a theorem of Steinberg on meromorphic families of compact operators. * Corresponding author 1991 Mathematics Subject Classification: 47A60, 47A05, 47A10

In this paper we define and study an extension of the g-Drazin for elements of a Banach algebra and for bounded linear operators based on an isolated spectral set rather than on an isolated spectral point. We investigate salient properties of the new inverse and its continuity, and illustrate its usefulness with an application to differential equations. Generalized Mbekhta subspaces are introduced and the corresponding extended Mbekhta decomposition gives a characterization of circularly isolated spectral sets.

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25–39) and of Cline and Greville (Linear Algebra Appl. 29 (1980), 53–62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakočević and Wei is solved.

Matrix theory and its applications make wide use of the eigenprojections of square matrices. The present paper demonstrates that the eigenprojection of a matrix A can be calculated with the use of any annihilating polynomial of A u , where u ≥ ind A. This enables one to find the components and the minimal polynomial of A, as well as the Drazin inverse A D .

Block representations of the Drazin inverse of a bipartite matrix A = ¾ 0 B C 0 ¿ in terms of the Drazin inverse of the smaller order block product BC or CB are presented. Relationships between the index of A and the index of BC are determined, and examples are given to illustrate all such possible relationships. http://math.technion.ac.il/iic/ela

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [A. Schuster, P. Hippe, Inversion of polynomial matrices by interpolation, IEEE Trans. Automat. Control 37 (3) (1992) 363–365], applicable to usual matrix inverse. Also, we improve our interpolation method, using a more effective estimation of degrees of polynomial matrices generated in Leverrier–Faddeev method. Algorithms are implemented and tested in the symbolic package MATHEMATICA.

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers and . and a continuation of the papers . The symbolic implementation of these algorithms in the package MATHEMATICA is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.