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 $A\{ 1,2,3,{ 1^{ k}}\}$.
Categories: math.NA cs.NA