Abstract: 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. |