Further results on the perturbation estimations for the Drazin inverse

Abstract

For n × n 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 E is small and the spectral radius ofThe acute perturbation coincides with the stable perturbation of the group inverse, if the matrix B satisfies geometrical condition:which introduced by Vélez-Cerrada, Robles, and Castro-González, (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).