A study on the 1-$\Gamma$ inverse of tensors via the M-Product

TL;DR

This paper explores the concept of the 1-$\Gamma$ inverse of tensors using the M-product, providing definitions, properties, algorithms, and solutions for related multilinear equations.

Contribution

It introduces the 1-$\Gamma$ inverse for tensors, establishes equivalent conditions, and develops numerical algorithms for computation and applications.

Findings

Defined and characterized the 1-$\Gamma$ inverse.

Developed algorithms based on singular value decomposition.

Validated results through numerical experiments.

Abstract

In this paper, we will study the issue about the 1-$\Gamma$ inverse, where $\Gamma\in\{\dag, D, *\}$, via the M-product. The aim of the current study is threefold. Firstly, the definition and characteristic of the 1-$\Gamma$ inverse is introduced. Equivalent conditions for a tensor to be a 1-$\Gamma$ inverse are established. Secondly, using the singular value decomposition, the corresponding numerical algorithms for computing the 1-$\Gamma$ inverse are given. Finally, the solutions of the multilinear equations related 1-$\Gamma$ inverse are studied, and numerical calculations are given to verify our conclusions.