Acute perturbation of the group inverse

For any n×n complex matrix A, let Ag be the group inverse of A. When A is singular, a matrix B=A+E is said to be an acute perturbation of A, if ‖E‖ is small and the spectral radius ρ(BgB−AgA)<1. The acute perturbation coincides with the stable perturbation of the group inverse, if the matrix B satisfies condition:R(B)∩N(A)={0},N(B)∩R(A)={0} which was introduced by Castro-González et al. (2008) [8]. Furthermore, several examples are provided to illustrate the acute perturbation of the group inverse.