The perturbation of the group inverse under the stable perturbation in a unital ring

TL;DR

This paper studies how the group inverse in a unital ring changes under stable perturbations, providing explicit formulas and extending previous results, with applications to matrix inverses.

Contribution

It derives explicit formulas for the perturbed group inverse in a unital ring under stable perturbations, extending prior theoretical results.

Findings

Explicit expressions for the perturbed group inverse are obtained.

The results generalize previous work in the literature.

Application to the group inverse of certain matrices is demonstrated.

Abstract

Let $\R $ be a ring with unit 1 and $a\in \R, \bar{a}=a+\delta a\in \R $ such that $a^#$ exists. In this paper, we mainly investigate the perturbation of the group inverse $a^#$ on $\R$. Under the stable perturbation, we obtain the explicit expressions of $\bar{a}^#$. The results extend the main results in Xue (2007), and Xue and Chen (2007) and some related results in Xue (2012). As an application, we give the representation of the group inverse of the matrix d&b c&0 on the ring $\R$ for certain $d, b, c\in\R$.