A $W$-weighted generalization of $\{1,2,3,1^{k}\}$-inverse for rectangular matrices (2312.01370v1)

Abstract: This paper presents a novel extension of the ${1,2,3,1^{k}}$-inverse concept to complex rectangular matrices, denoted as a $W$-weighted ${1,2,3,1^{k}}$-inverse (or ${1',2',3',{1^{k}}'}$-inverse), where the weight $W \in \mathbb{C}^{n \times m}$. The study begins by introducing a weighted ${1,2,3}$-inverse (or ${1',2',3'}$-inverse) along with its representations and characterizations. The paper establishes criteria for the existence of ${1',2',3'}$-inverses and extends the criteria to ${1'}$-inverses. It is further demonstrated that $A\in \mathbb{C}^{m \times n}$ admits a ${1',2',3',{1^{k}}'}$-inverse if and only if $r(WAW)=r(A)$, where $r(\cdot)$ is the rank of a matrix. The work additionally establishes various representations for the set $A{ 1',2',3',{1^{k}}'}$, including canonical representations derived through singular value and core-nilpotent decompositions. This, in turn, yields distinctive canonical representations for the set $A{ 1,2,3,{1^{k}}}$. ${ 1',2',3',{1^{k}}'}$-inverse is shown to be unique if and only if it has index $0$ or $1$, reducing it to the weighted core inverse. Moreover, the paper investigates properties and characterizations of ${1',2',3',{1^{k}}'}$-inverses, which then results in new insights into the characterizations of the set $A{ 1,2,3,{1^{k}}}$.