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 …