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 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].     

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.     

The Pseudo Drazin inverses in Banach Algebras

Let (A) be a complex Banach algebra and J be the Jacobson radical of(A).(1) We firstly show that a is generalized Drazin invertible in (A) if and only if a+J is generalized Drazin invertible in (A)/J.Then we prove that a is pseudo Drazin invertible in (A) if and only if a + J is Drazin invertible in (A)/J.As its application,the pseudo Drazin invertibility of elements in a Banach algebra is explored.(2) The pseudo Drazin order is introduced in (A).We give the necessary and sufficient conditions under which elements in (A) have pseudo Drazin order,then we prove that the pseudo Drazin order is a pre-order.     

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.     

Expression of the Drazin and MP-inverse of partitioned matrix and quotient identity of generalized Schur complement

In this paper we give representations of the Drazin and MP-inverse of a 2×2 block matrix and quotient identities for the generalized Schur complement of a partitioned 3×3 matrix under conditions different than those used in recent papers on the subject. We present numerical examples to illustrate our results.     

On the Drazin inverses involving power commutativity

We explore the Drazin inverses of bounded linear operators with power commutativity (PQ=QmP) in a Hilbert space. Conditions on Drazin invertibility are formulated and shown to depend on spectral properties of the operators involved. Moreover, we prove that P±Q is Drazin invertible if P and Q are dual power commutative (PQ=QmP and QP=PnQ) and show that the explicit representations of the Drazin inverse D(P±Q) depend on the positive integers m,n?2.     

Some results on the Drazin inverse of anti-triangular matrices

Abstract

The representations for the Drazin inverse of anti-triangular matrices are obtained under some conditions. Applying these representations, we give a necessary condition for a class of block matrices to have signed Drazin inverse.     

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].     

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.     

The Drazin inverses of products and differences of orthogonal projections

In this note, the Drazin inverses of products and differences of orthogonal projections on a Hilbert space are established.     

A note on the Drazin inverses with Banachiewicz-Schur forms

Let H be a Hilbert space, M the closed subspace of H with orthocomplement M^⊥. According to the orthogonal decomposition H=M⊕M^⊥, every operator M∈B(H) can be written in a block-form . In this note, we give necessary and sufficient conditions for a partitioned operator matrix M to have the Drazin inverse with Banachiewicz-Schur form. In addition, this paper investigates the relations among the Drazin inverse, the Moore-Penrose inverse and the group inverse when they can be expressed in the Banachiewicz-Schur forms.     

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.