Greville type {1,2,3}-generalized inverses for rectangular matrices

Abstract

For any complex matrices $A$ and $W$, $m \times n$ and $n \times m$, respectively, it is proved that there exists a complex matrix $X$ such that $AXA=A$, $XAX=X$, $(AX)^{\ast}=AX$ and $XA(WA)^{k}=(WA)^{k}$, where $k$ is the index of $WA$. When $A$ is square and $W$ is the identity matrix, such an $X$ reduces to Greville's spectral $\{1,2,3\}$-inverse of $A$. Various expressions of such generalized inverses are established.