Characterizations of a Class of Matrices and Perturbation of the Drazin Inverse

Given a singular square matrix A with index r, ind(A) = r, we establish several characterizations in the Drazin inverse framework of the class of matrices B, which satisfy the conditions N (Bs)∩R(Ar) = {0} and R(Bs)∩N (Ar) = {0} with ind(B) = s, where N (A) and R(A) denote the null space and the range space of a matrix A, respectively. We give explicit representations for BD and BBD and upper bounds for the errors ‖BD − AD‖/‖AD‖ and ‖BBD − AAD‖. In a numerical example we show that our bounds are better than others given in the literature.