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Characterizations and perturbation bounds for the Drazin inverse of a class of matrices

Resumo: Campbell and Meyer established the following continuity property for the Drazin inverse: Given a singular square matrix $A$ with index $r$, if $A_j\rightarrow A$, then $A_j^{\rm D}\rightarrow A^{\rm D}$ if and only if $\rank A_j^{r_j} = \rank A^r $ for all sufficiently large $j$, where $r_j$ is the index of $A_j$. An open  problem is to obtain explicit upper bounds for the errors $\norm{B^{D}-A^D}/\norm{A^D}$ under condition $\rank{B^s}=\rank{A^r}$, where $s$ is the index of $B$. In this talk we will give several characterizations in the Drazin inverse framework of  the class of  matrices $B$ which satisfy the conditions $\nul{B^s} \cap \ran{A^r}=\{ 0\}$ and $\ran{B^s} \cap \nul{A^r} =\{0\}$, where $\nul{A}$ and $\ran{A}$ denote the null space and the range space of a matrix $A$, respectively.
We will see that the above matrices can be characterized by the rank matrix conditions $\rank{A^r}=\rank{B^s}=\rank{A^rB^s}=\rank{B^s A^r}$ and we will give explicit representations for $B^{D}$ and $BB^{D}$ and upper bounds for the errors $\norm{B^{D}-A^D}/\norm{A^D}$ and $\norm{BB^{D}- AA^D}$. In a numerical example we show that our bounds are better than others given in the literature.
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