The perturbation of Drazin inverse and dual Drazin inverse
Hongxing Wang, Chong Cui, Yimin Wei · Special Matrices · 2024
Abstract In this study, we derive the Drazin inverse (A+εB)D {\left(A+\varepsilon B)}^{D} of the complex matrix A+εB A+\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+εB A+\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 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+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.