Jacobson’s lemma and Cline’s formula for minimal rank weak Drazin inverses

Abstract

Minimal rank weak Drazin inverses of a square complex matrix $M$ are solutions to the matrix equation $XM^{k+1}=M^{k}$ with the minimal rank, covering a large class of generalized inverses. In this paper, minimal rank weak Drazin inverses of $I_n-BA$ (resp. $BA$) are expressed in terms of that of $I_m-AB$ (resp. $AB$), where $A$ and $B$ are $m \times n$ and $n \times m$ complex matrices, respectively.