A study on the 1-$Γ$ inverse of tensors via the M-Product
Siran Chen, Hongwei Jin, Shaowu Huang, Julio Benítez
TL;DR
This work extends the M-product framework by introducing and characterizing the $1-Γ$ inverse, with $Γ ∈ \{†, D, *\}$, and developing practical SVD-based methods to compute the three variants: $1$-MP, $1$-D, and $1$-Star inverses. It provides explicit tensor decompositions, nonuniqueness characterizations, and algorithmic procedures to obtain these inverses via frontal-slice computations and mode-3 transforms. The authors also derive solution forms for multilinear systems using each inverse, detailing the corresponding general and particular solutions and demonstrating how these inverses organize solution sets. The results enable robust tensor inversion and system-solving in the $M$-product setting, complemented by an example program and applications to multilinear equations. Overall, the paper offers a unified, computationally tractable approach to 1-Γ inverses and their use in multilinear problem contexts.
Abstract
In this paper, we will study the issue about the 1-$Γ$ inverse, where $Γ\in\{†, D, *\}$, via the M-product. The aim of the current study is threefold. Firstly, the definition and characteristic of the 1-$Γ$ inverse is introduced. Equivalent conditions for a tensor to be a 1-$Γ$ inverse are established. Secondly, using the singular value decomposition, the corresponding numerical algorithms for computing the 1-$Γ$ inverse are given. Finally, the solutions of the multilinear equations related 1-$Γ$ inverse are studied, and numerical calculations are given to verify our conclusions.