The Perturbation of the Group Inverse Under the Stable Perturbation in a Unital Ring

Fapeng Du, Yifeng Xue

Let $\frak R$ be a ring with unit 1 and $a\in\frak R$, $\bar a=a+\delta a\in\frak R$ such that $a^{\#}$ exists. In this paper, we mainly investigate the perturbation of the group inverse $a^{\#}$ on $\frak R$. Under the stable perturbation, we obtain the explicit expressions of $\bar a^{\#}$. The results extend the main results in [19,20] and some related results in [18]. As an application, we give the representation of the group inverse of the matrix $\big[\smallmatrix d&b\\c&0\endsmalmatrix\bigr]$ on the ring $\frak R$ for certain $d, b, c \in \frak R$.