• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.113-126
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• 2007

Let A and E be $m{\times}n$ matrices and W an $n{\times}m$ matrix, and let $A_{d,w}$ denote the W-weighted Drazin inverse of A. In this paper, a new representation of the W-weighted Drazin inverse of A is given. Some new properties for the W-weighted Drazin inverse $A_{d,w}\;and\;B_{d,w}$ are investigated, where B=A+E. In addition, the Banach-type perturbation theorem for the W-weighted Drazin inverse of A and B are established, and the perturbation bounds for ${\parallel}B_{d,w}{\parallel}\;and\;{\parallel}B_{d,w}-A_{d,w}{\parallel}/{\parallel}A_{d,w}{\parallel}$ are also presented. When A and B are square matrices and W is identity matrix, some known results in the literature related to the Drazin inverse and the group inverse are directly reduced by the results in this paper as special cases.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.765-771
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• 2014

Necessary and sufficient conditions for the existence of the group inverse of an anti-triangular block matrix in Banach algebras are presented. Using these results, we give the formulae for the generalized Drazin inverse of a block matrix.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.267-273
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• 2009

A representation for the Drazin inverse of an arbitrary square matrix in terms of the eigenprojection was established by Rothblum [SIAM J. Appl. Math., 31(1976) :646-648]. In this paper perturbation results based on the representation for the Darzin inverse $A^D\;=\;(A-X)^{-1}(I-X)$ are developed. Norm estimates of $\parallel(A+E)^D-A^D\parallel_2/\parallel A^D\parallel_2$ and $\parallel(A+E)^#-A^D\parallel_2/\parallel A^D\parallel_2$ are derived when IIEI12 is small.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.705-718
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• 2022

In this paper, for bounded linear operators A, B, C satisfying [AB, B] = [BC, B] = [AB, BC] = 0 we study the Drazin invertibility of the sum of products formed by the three operators A, B and C. In particular, we give an explicit representation of the anti-commutator {A, B} = AB + BA. Also we give some conditions for which the sum A + C is Drazin invertible.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.35-59
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• 2006

In this paper, we generalized the results of [23, 26], and get the results of the condition number of the W-weighted Drazin-inverse solution of linear system W AW\chi=b, where A is an $m{\times}n$ rank-deficient matrix and the index of A W is $k_1$, the index of W A is $k_2$, b is a real vector of size n in the range of $(WA)^{k_2}$, $\chi$ is a real vector of size m in the range of $(AW)^{k_1}$. Let $\alpha$ and $\beta$ be two positive real numbers, when we consider the weighted Frobenius norm $\|[{\alpha}W\;AW,\;{\beta}b]\|$(equation omitted) on the data we get the formula of condition number of the W-weighted Drazin-inverse solution of linear system. For the normwise condition number, the sensitivity of the relative condition number itself is studied, and the componentwise perturbation is also investigated.

• Kyungpook Mathematical Journal
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• v.64 no.2
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• pp.205-218
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• 2024

Let 𝒜 be a Banach algebra. An element a ∈ 𝒜 has generalized Drazin inverse if there exists b ∈ 𝒜 such that b = bab, ab = ba, a - a²b ∈ 𝒜^qnil. New additive results for the generalized Drazin inverse of an operator over a Banach space are presented. we extend the main results of a paper of Shakoor, Yang and Ali from 2013 and of Wang, Huang and Chen from 2017. Appling these results to 2×2 operator matrices we also generalize results of a paper of Deng, Cvetković-Ilić and Wei from 2010.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1263-1271
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• 2018

We introduce new generalized inverses (named the WgDMP inverse and dual WgDMP inverse) for a bounded linear operator between two Hilbert spaces, using its Wg-Drazin inverse and its Moore-Penrose inverse. Some new properties of WgDMP inverse and dual WgDMP inverse are obtained and some known results are generalized.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.81-94
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• 2007

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.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.541-547
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• 2008

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with $n\;{\geq}\;3$. We denote by $M_n(R)$ the ring of all $n{\times}n$ matrices over R. Let ($J_n(R)$) be the additive subgroup of $M_n(R)$ generated additively by all idempotent matrices. Let ($D=J_n(R)$) or $M_n(R)$. In this paper, by using an additive idem potence-preserving result obtained by Coo (see [4]), I characterize (i) the additive preservers of tripotence from D to $M_m(R)$ when 2 and 3 are units of R; (ii) the additive preservers of inverses (respectively, Drazin inverses, group inverses, {1}-inverses, {2}-inverses, {1, 2}-inverses) from $M_n(R)$ to $M_n(R)$ when 2 and 3 are units of R.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.1-16
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• 2006

The classical perturbation theory is extended to the weighted Kronecker product linear systems $W(A{\bigotimes}B)W\chi=h$. Upper bounds are derived for the normwise condition number.