Acute perturbation of Drazin inverse and oblique projectors

For an n×n complex matrix A with ind(A) = r; let AD and Aπ = I–AAD be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) = s, it is said to be a stable perturbation of A, if I–(Bπ–Aπ)2 is nonsingular, equivalently, if the matrix B satisfies the condition R(Bs)∩N(Ar)={0}$$\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\left\{0\right\}$$ and N(Bs)∩R(Ar)={0}$$\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\left\{0\right\}$$, introduced by Castro-González, Robles, and Vélez-Cerrada. In this paper, we call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius ρ(Bπ–Aπ) < 1: We present a perturbation analysis and give suffcient and necessary conditions for a perturbation of a square matrix being acute with respect to the matrix Drazin inverse. Also, we generalize our perturbation analysis to oblique projectors. In our analysis, the spectral radius, instead of the usual spectral norm, is used. Our results include the previous results on the Drazin inverse and the group inverse as special cases and are consistent with the previous work on the spectral projections and the Moore-Penrose inverse.