Generalized Drazin inverses of anti-triangular block matrices

In this paper we give formulae for the generalized Drazin inverse Md of an anti-triangular matrix in two different ways: one is to express Md in terms of Ad with arbitrary B and C, the other is to express Md in terms of Bd and Cd with arbitrary A. Moreover, the results are applied to obtain generalized Drazin inverses of various structured matrices and some special cases are analyzed.     

A note on the Drazin inverse of an anti-triangular matrix

In this paper we give formulae for the generalized Drazin inverse M^d of an anti-triangular matrix M under some conditions. Moreover, some particular cases of these results are also considered.     

The representations of the Drazin inverse of differences of two matrices

In this paper, we give explicit representations of (P ± Q)^D, (P ± PQ)^D and (PQ)^# of two matrices P and Q, as a function of P, Q, P^D and Q^D, under the conditions P³Q = QP and Q³P = PQ.     

Some additive results on Drazin inverses

In this paper, some additive results on Drazin inverse of a sum of Drazin invertible elements are derived. Some converse results are also presented.     

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.     

A note on the representation for the Drazin inverse of block matrices

In this note we consider representations of the Drazin inverse of 2×2 block matrices under conditions weaker than those used in recent papers on the subject, in particular in [D.S. Djordjević, P.S. Stanimirović, On the generalized Drazin inverse and generalized resolvent, Czechoslovak Math. J. 51 (126) (2001) 617–634; R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (2006) 757–771].     

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.     

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.     

On the Drazin inverse of block matrices and generalized Schur complement

In this paper, we give an additive result for the Drazin inverse with its applications, we obtain representations for the Drazin inverse of a 2 × 2 complex block matrix having generalized Schur complement S=D-CA^DB equal to zero or nonsingular. Several situations are analyzed and recent results are generalized [R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (3) (2006) 757-771].     

Characterizations and representations of the Drazin inverse involving idempotents

This paper is to present some results on the Drazin invertibility of products and differences of idempotents. In addition, some formulae for the Drazin inverse of sums, differences and products of idempotents are also established.     

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.     

A note on the perturbation bound of the Drazin inverse

A perturbation bound for the Drazin inverse A^D with Ind(A+E)=1 has recently been developed. However, those upper bounds are not satisfied since it is not tight enough. In this paper, a sharper upper bounds for ||(A+E)^#−A^D|| with weaker conditions is derived. That new bound is also a generalization of a new general upper bound of the group inverse. We also derive a new expression of the Drazin inverse (A+E)^D with Ind(A+E)>1 and the corresponding upper bound of ||(A+E)^D−A^D|| in a special case. Numerical examples are given to illustrate the sharpness of the new bounds.     

Representations and properties of the W-Weighted Drazin inverse

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.     

New additive results on Drazin inverse and its applications

In this paper, the Drazin inverse of certain sums and block triangular matrices are presented.     

An oblique projection iterative method to compute Drazin inverse and group inverse

In this paper, an oblique projection iterative method is presented to compute matrix equation AXA=A of a square matrix A with ind(A)=1. By this iterative method, when taken the initial matrix X₀=A, the group inverse A_g can be obtained in absence of the roundoff errors. If we use this iterative method to the matrix equation A^kXA^k=A^k, a group inverse (A^k)_g of matrix A^k is got, then we use the formulae A_d=A^k-1(A^k)_g, the Drazin inverse A_d can be obtained.