Further results on generalized inverses of tensors via the Einstein product

TL;DR

This paper advances the theory of Moore-Penrose inverses for tensors with Einstein product, providing new characterizations, computational methods, and reverse order laws, and applying these to solve multilinear tensor systems.

Contribution

It introduces new characterizations and a novel method for computing tensor Moore-Penrose inverses, expanding the theoretical framework and practical applications.

Findings

New characterizations of tensor generalized inverses

A novel method for computing Moore-Penrose inverse of tensors

Reverse order laws for tensor inverses

Abstract

The notion of the Moore-Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore-Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.