The perturbation of Drazin inverse and dual Drazin inverse

In this study, we derive the Drazin inverse (A+εB)D{\left(A+\varepsilon B)}^{D} of the complex matrix A+εBA+\varepsilon B with Ind(A+εB)>1{\rm{Ind}}\left(A+\varepsilon B)\gt 1 and Ind(A)=k{\rm{Ind}}\left(A)=k and the group inverse (A+εB)#{\left(A+\varepsilon B)}^{\#} of the complex matrix A+εBA+\varepsilon B with Ind(A+εB)=1{\rm{Ind}}\left(A+\varepsilon B)=1 and Ind(A)=k{\rm{Ind}}\left(A)=k when εB\varepsilon B is viewed as the perturbation of AA. If the dual Drazin inverse (DDGI) A^DDGI{\widehat{A}}^{{\rm{DDGI}}} of A^\widehat{A} is considered as a notation. We calculate (A+εB)D−A^DDGI{\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}} and (A+εB)#−A^DDGI{\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}} and obtain ‖(A+εB)D−A^DDGI‖P∈O(ε2)\Vert {\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2}) and ‖(A+εB)#−A^DDGI‖P∈O(ε2)\Vert {\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2}). Meanwhile, we give some examples to verify these conclusions.