Further Results on the Drazin Inverse of Tensors via Einstein Product

In this paper,we give further results on the Drazin inverse of tensors via the Einstein product.We give a limit formula for the Drazin inverse of tensors.By using this formula,the representations for the Drazin inverse of several block tensor are obtained.Further,we give the Drazin inverse of the sum of two tensors based on the representation for the Drazin inverse of a block tensor.     

Jacobson’s lemma for the generalized Drazin inverse

0truemm0truemm We study properties of elements in a ring which admit the generalized Drazin inverse. It is shown that the element 1-ab is generalized Drazin invertible if and only if so is 1-ba and a formula for the generalized Drazin inverse of 1-ba in terms of the generalized Drazin inverse and the spectral idempotent of 1-ab is provided. Further, recent results relating to the Drazin index can be recovered from our theorems.     

分块态射的广义逆

ClineRE给出了分块矩阵的Moore-Penrose逆的表达式,PetrPeska引进了分块态射的记号且导出了分块态射的Moore-Penrose逆的表达式.本文中,我们推广了Cline型分块态射的记号并得到了Cline型分块态射的Moore-Penrose逆和Drazin逆以及群逆的表达式.     

Some additive results on Drazin inverse

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.     

A generalization of the Sherman–Morrison–Woodbury formula

In this paper, we develop conditions under which the Sherman–Morrison–Woodbury formula can be represented in the Moore–Penrose inverse and the generalized Drazin inverse forms. These results generalize the original Sherman–Morrison–Woodbury formula.     

Formulae for the generalized Drazin inverse of a block matrix in terms of Banachiewicz–Schur forms

We introduce new expressions for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra under some conditions. We generalized some recent results for Drazin inverse and group inverse of complex matrices.     

态射的Drazin逆

本文研究范畴中态射的Ｄｒａｚｉｎ逆．给出了一般范畴中态射的｛１ｍ，２，５｝逆的一个等价刻划．在Ａｂｅｌ范畴中，建立指数与Ｄｒａｚｉｎ逆的概念，证明了有Ｄｒａｚｉｎ逆的态射必有柱心－幂零分解．     

The Drazin Inverse of a Morphism with a Sequence of (epic, monic) Factorizations

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎ [1 ],ｔｈｅａｕｔｈｏｒｄｅｆｉｎｅｄａｃｃｕｒａｔｅｌｙｔｈｅＤｒａｚｉｎｉｎｖｅｒｓｅｏｆａｍｏｒｐｈｉｓｍ .Ｐｒｅｖｉｏｕｓｌｙ ,ｓｏｍｅｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｓｗｅｒｅｏｂｔａｉｎｅｄｆｏｒａｍｏｒｐｈｉｓｍｗｉｔｈ (ｅｐｉｃ ,ｍｏｎｉｃ)ｆａｃｔｏｒｉｚａｔｉｏｎｔｏｈａｖｅｔｈｅｇｒｏｕｐｉｎｖｅｒｓｅ (ｓｅｅ [2 ],Ｔｈｅｏｒｅｍ 1 )ａｎｄｔｈｅＤｒａｚｉｎｉｎｖｅｒｓｅｏｆａｍａｔｒｉｘｏｖｅｒｔｈｅｃｏｍｐ…     

The Drazin inverse in an arbitrary semiring

In this article, we investigate the Drazin invertibility for the elements of an arbitrary semiring. We give necessary and sufficient conditions for the existence and expressions of the Drazin inverse of an element in an arbitrary semiring. Moreover, we consider the product paq under some additional necessary conditions for which the Drazin inverse of the product paq exists.     

A note on computational formulas for the Drazin inverse of certain block matrices

In this paper, we give a computational formula for the Drazin inverse of the sum P+Q, then applying it we give some computational formulas for the Drazin inverse of block matrix (A and D are square) with generalized Schur complement S=D?CA ^D B is nonsingular under some conditions. These results extend the results about the Drazin inverse of M given by R. Hartwig, X. Li and Y.?Wei (SIAM J. Matrix Anal. Appl. 27:757?C771, 2006) and by C. Deng (J. Math. Anal. Appl. 368:1?C8, 2010).     

Nonnegative Drazin inverses

A characterization of nonnegative matrices which have a nonnegative Drazin inverse is given. A necessary and sufficient condition for a real matrix to have a nonnegative Drazin inverse is also presented.     

A finite algorithm for the Drazin inverse of a polynomial matrix

Based on Greville's finite algorithm for Drazin inverse of a constant matrix we propose a finite numerical algorithm for the Drazin inverse of polynomial matrices. We also present a new proof for Decell's finite algorithm through Greville's finite algorithm.     

Computing generalized inverses of a rational matrix and applications

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 [11] and [14]. and a continuation of the papers [15, 16]. 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.     

A weighted Drazin inverse and applications

In this paper the notation of the Cline–Greville W-weighted Drazin inverse of a rectangular matrix is extended to bounded linear operators between Banach spaces. We give new characterizations of the W-weighted Drazin inverse, and we study the perturbations and the the continuity of the W-weighted Drazin inverse.     

Challenging Problems on the Perturbation of Drazin Inverse

In order to estimate error bounds on the computed Drazin inverse of a matrix, we need to establish some perturbation theory for the Drazin inverse which is analogous to that for the Moore–Penrose inverse. In this paper, we present recent results on this topic, three problems are put forward in this direction.     

Representation for the W-weighted Drazin inverse of linear operators

In this paper we study the W-weighted Drazin inverse of the bounded linear operators between Banach spaces and its representation theorem. Based on this representation, utilizing the spectral theory of Banach space operators, we derive an approximating expression of the W-weighted Drazin inverse and an error bound. Also, a perturbation theorem for the W-weighted Drazin inverse is uniformly obtained from the representation theorem.     

关于带W权Drazin逆的表示定理及迭代方法

To study singular linear system, Cline and Greville[8] proposed the concept of W-weighted Drazin inverse for the rectangular matrices,where the properties were also discussed. The computation for the W-weighted Drazin inverse is of much interest, which is mainly divided into two kinds of methods: direct method[2,4,6] and iterative method[3,5,7,9,12,13]. In this paper, we study the iterative method and successive matrix squaring(SMS) method for the W-weighted Drazin inverse and generalize the main results in [12,13].     

Additive results for the Drazin inverse of block matrices and applications

In this paper, we consider the Drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject. As a corollary we get the main results from the paper of Yang and Liu [H. Yang, X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math. 235 (2011) 1412-1417]. As an application we give some new representations for the Drazin inverse of a block matrix.     

Variation of the parameters formula and the problem of BIBO for singular measure differential systems with impulse effect

Recently, we first presented and studied singular measure differential systems with impulsive solutions. Some fundamental theory was established. In the present paper, with the help of the Drazin inverse, the variation of the parameters formula of singular measure differential systems is obtained. As an application, the problem of boundary input and boundary output (BIBO) is investigated.     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.