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.     

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.     

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.     

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.     

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.     

On the Generalized Drazin Inverse and Generalized Resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C ^*-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.     

On Drazin inverse of operator matrices

In this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions.     

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.     

布尔矩阵的Cline逆

研究了布尔矩阵的广义逆,首先引入了布尔矩阵的Drazin逆及Cline逆,利用布尔矩阵的性质证明了任意布尔矩阵均有Drazin逆,从而证得任意布尔矩阵均有Cline逆,且Cline唯一.而且,在A+存在的情况下Ac=A+.最后证明了Cline逆的一些性质.     

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.     

The (2,2,0) Drazin inverse problem

We consider the additive Drazin problem and we study the existence of the Drazin inverse of a two by two matrix with zero (2,2) entry.     

A note on the perturbation bounds of W-weighted Drazin inverse

We investigate the perturbation bound for the W-weighted Drazin inverse of a rectangular matrix and present two explicit expressions for the W-weighted Drazin inverse under the one-sided condition, which extends the results in Appl. Math. Comput. 2004;149:423–430.     

Error Bounds for Perturbation of the Drazin Inverse of Closed Operators with Equal Spectral Projections

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, Rako ) evi ' 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.     

A NOTE ON THE REPRESENTATIONS FOR THE GENERALIZED DRAZIN INVERSE OF BLOCK MATRICES

We present some representations for the generalized Drazin inverse of a block matrix x =[abcd]in a Banach algebra A in terms of a~d and(bc)~d under certain conditions,extending some recent result related to the generalized Drazin inverse of an anti-triangular operator matrix.Also,several particular cases of this result are considered.     

A note on additive results for the Drazin inverse

For two square matrices that commute, we present some additive results for the Drazin inverse. We also give the application to relative perturbation of eigenvalues when the perturbed matrix commutes with the original matrix and perturbation bounds of the Drazin inverse.     

Further results on the Drazin inverse of even‐order tensors

The notion of the Drazin inverse of an even‐order tensors with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402‐3413]. In this article, we further elaborate this theory by establishing a few characterizations of the Drazin inverse and $\mathrm{??}$‐weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution of multilinear systems by using the Drazin inverse and iterative (higher order Gauss‐Seidel) method of tensors. Besides these, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product.     

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．     

New additive results for the generalized Drazin inverse

In this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of PQ=QP, we give explicit representations of the generalized Drazin inverse d(P+Q) in term of P, Pd, Q and Qd. We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.     

The Drazin inverse as a gradient

The coefficients in the expansion of adj(λI ? A) are expressed as gradients, and some new representations are given for the Drazin inverse of a matrix over an arbitrary field. These results are then combined to express the Drazin inverse as a gradient of a function of the entries of the matrix.