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

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 $\{1^{\mathrm{\prime }},2^{\mathrm{\prime }},3^{\mathrm{\prime }},{1^k}^{\mathrm{\prime }}\}$-inverse), where the weight $W\in {\mathbb{C}}^{n\times m}$. The study begins by introducing a weighted $\{1,2,3\}$-inverse (or $\{1^{\mathrm{\prime }},2^{\mathrm{\prime }},3^{\mathrm{\prime }}\}$-inverse) along with its representations and characterizations. The paper establishes criteria for the existence of $\{1^{\mathrm{\prime }},2^{\mathrm{\prime }},3^{\mathrm{\prime }}\}$-inverses and extends the criteria to $\{1^{\mathrm{\prime }}\}$-inverses. It is further demonstrated that $A\in {\mathbb{C}}^{m\times n}$ admits a $\{1^{\mathrm{\prime }},2^{\mathrm{\prime }},3^{\mathrm{\prime }},{1^k}^{\mathrm{\prime }}\}$-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 $A\{1^{\mathrm{\prime }},2^{\mathrm{\prime }},3^{\mathrm{\prime }},{1^k}^{\mathrm{\prime }}\}$, including canonical representations derived through singular value and core-nilpotent decompositions. This, in turn, yields distinctive canonical representations for the set $A\{1,2,3,1^k\}$. $\{1^{\mathrm{\prime }},2^{\mathrm{\prime }},3^{\mathrm{\prime }},{1^k}^{\mathrm{\prime }}\}$-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 $\{1^{\mathrm{\prime }},2^{\mathrm{\prime }},3^{\mathrm{\prime }},{1^k}^{\mathrm{\prime }}\}$-inverses, which then results in new insights into the characterizations of the set $A\{1,2,3,1^k\}$.