The perturbation of the group inverse under the stable perturbation in a unital ring
Provides theoretical extensions to existing results on group inverse perturbations in ring theory, but is incremental in nature.
This paper investigates the perturbation of the group inverse under stable perturbations in a unital ring, deriving explicit expressions for the perturbed group inverse. The results extend previous work by Xue (2007) and Xue and Chen (2007).
Let $\R $ be a ring with unit 1 and $a\in \R, \bar{a}=a+δa\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 $d, b, c\in\R$.