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. |