FURTHER RESULTS ON THE PERTURBATION ESTIMATIONS FOR THE DRAZIN INVERSE

For nxn complex singular matrix A with ind(A) = k > 1, let A(D) be the Drazin inverse of A. If a matrix B = A + E with ind(B) = 1 is said to be an acute perturbation of A, if parallel to Ek parallel to is small and the spectral radius of BgB-A(D)A satisfies rho(BgB-A(D)A) < 1, where B-g is the group inverse of B. The acute perturbation coincides with the stable perturbation of the group inverse, if the matrix B satisfies geometrical condition: R(B) boolean AND N(A(k)) = {0}, N(B) boolean AND R(A(k)) = {0} which introduced by Velez-Cerrada, Robles, and Castro-Gonzalez, (Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput., 215 (2009), 2154-2161). Furthermore, two examples are provided to illustrate the acute perturbation of the Drazin inverse. We prove the correctness of the conjecture in a special case of ind(B) = 1 by Wei (Acute perturbation of the group inverse, Linear Algebra Appl., 534 (2017), 135-157).