The perturbation of Drazin inverse and dual Drazin inverse

In this study, we derive the Drazin inverse (A + epsilon B)(D) of the complex matrix A + epsilon Bwith Ind(A+ epsilon B) > 1 and Ind(A) = k and the group inverse (A + epsilon B)(# )of the complex matrix A + epsilon Bwith Ind(A + epsilon B) = 1 and Ind(A) = k when epsilon B is viewed as the perturbation of A. If the dual Drazin inverse (DDGI) A(<^>DDGI) of A(<^>) is considered as a notation. We calculate (A + epsilon B)(D) - A(<^>DDGI) and (A + epsilon B)(# ) - A(<^>DDGI) and obtain & Vert;(A+ epsilon B)(D )- A(<^>DDGI )& Vert;(P )is an element of O(epsilon(2)) and & Vert;(A+ epsilon B)(#)- A(<^>DDGI)& Vert;(P) is an element of O(epsilon(2)). Meanwhile, we give some examples P is an element of epsilon to verify these conclusions.