An extension of the Moore-Penrose Inverse of a Tensor via the Einstein product

TL;DR

This paper extends the concept of the Moore-Penrose inverse to tensors using the Einstein product, introduces a new generalized inverse, and explores conditions for their equivalence and properties like the reverse order law.

Contribution

It proposes a new tensor inverse called the product Moore-Penrose inverse and establishes conditions for its relation to the standard Moore-Penrose inverse.

Findings

Derived an expression for the Moore-Penrose inverse of tensor products

Introduced the product Moore-Penrose inverse for tensors

Established conditions for inverse coincidence and the reverse order law

Abstract

In this paper, we first give an expression for the Moore-Penrose inverse of the product of two tensors via the Einstein product. We then introduce a new generalized inverse of a tensor called product Moore-Penrose inverse. A necessary and sufficient condition for the coincidence of the Moore-Penrose inverse and the product Moore-Penrose inverse is also proposed. Finally, the triple reverse order law of tensors is introduced.