The DMP Inverse for Rectangular Matrices

Abstract

The definition of the DMP inverse of a square matrix with complex elements is extended to rectangular matrices by showing that for any $A$ and $W$, $m$ by $n$ and $n$ by $m$, respectively, there exists a unique matrix $X$, such that
$$
XAX=X,~XA=WA_{d,w}WA~~\mbox{and}~~(WA)^{k+1}X=(WA)^{k+1}A^\dag,
$$
where $A_{d,w}$ denotes the $W$-weighted Drazin inverse of $A$ and $k=\mbox{Ind}(AW)$, the index of $AW$.