The perturbation of Drazin inverse and dual Drazin inverse

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