Further results on the perturbation estimations for the Drazin inverse

Haifeng Ma; Xiaoshuang Gao

doi:10.3934/naco.2018031

Article Contents

2018, Volume 8, Issue 4: 493-503. Doi: 10.3934/naco.2018031

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Further results on the perturbation estimations for the Drazin inverse

Haifeng Ma^, and
Xiaoshuang Gao

School of Mathematical Science, Harbin Normal University, Harbin 150025, China

* Corresponding author: H. Ma
* Corresponding author: H. Ma

Received: February 2018

Revised: March 2018

Published: September 2018

H. Ma is supported by Scientific Research Foundation of Heilongjiang Provincial Education Department (Grant No.12541232)
X. Gao is supported by Scientific Research Foundation of Heilongjiang Provincial Education Department (Grant No.12541232).

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Abstract

For $n× n$ complex singular matrix $A$ with ind$(A) = k>1$, let $A^D$ be the Drazin inverse of $A$. If a matrix $B = A+E$ with ind$(B) = 1$ is said to be an acute perturbation of $A$ , if $\|E\|$ is small and the spectral radius of $B_gB- A^DA$ satisfies

$ρ(B_gB- A^DA) < 1,$

where $B_g$ is the group inverse of $B$ .

The acute perturbation coincides with the stable perturbation of the group inverse, if the matrix $B$ satisfies geometrical condition:

${\mathcal R}(B) \cap {\mathcal N}(A^k) = \{ {\bf 0} \}, {\mathcal N}(B)\cap {\mathcal R}(A^k) = \{ {\bf 0} \}$

which introduced by Vélez-Cerrada, Robles, and Castro-González, (Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput., 215 (2009), 2154-2161).

Furthermore, two examples are provided to illustrate the acute perturbation of the Drazin inverse. We prove the correctness of the conjecture in a special case of ind$(B) = 1$ by Wei (Acute perturbation of the group inverse, Linear Algebra Appl., 534 (2017), 135-157).

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Mathematics Subject Classification: Primary: 15A09; Secondary: 65F20.

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