Characterizations of a Class of Matrices and Perturbation of the Drazin Inverse
N. Castro-González, J. Robles, J.Y. Vélez-Cerrada · SIAM Journal on Matrix Analysis and Applications · 2008
Given a singular square matrix A with index r, $\operatorname{ind}(A)=r$, we establish several characterizations in the Drazin inverse framework of the class of matrices B, which satisfy the conditions $\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\{0\}$ and $\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\{0\}$ with $\operatorname{ind}(B)=s$, where $\mathcal{N}(A)$ and $\mathcal{R}(A)$ denote the null space and the range space of a matrix A, respectively. We give explicit representations for $B^{\rm D}$ and $BB^{\rm D}$ and upper bounds for the errors $\|B^{\rm D}-A^{\rm D}\|/\|A^{\rm D}\|$ and $\|BB^{\rm D}-AA^{\rm D}\|$. In a numerical example we show that our bounds are better than others given in the literature.