# Weighted Moore-Penrose inverses of arbitrary-order tensors

## TL;DR

This paper investigates the properties and representations of weighted Moore-Penrose inverses of arbitrary-order tensors using Einstein product, including their decompositions, cancellation properties, and conditions for reverse-order law.

## Contribution

It introduces a new algebraic approach to characterize and compute weighted Moore-Penrose inverses of tensors, extending existing tensor inverse theory.

## Key findings

- Derived singular value and full-rank decompositions for tensors.
- Established properties and representations of weighted Moore-Penrose inverses.
- Provided conditions for the reverse-order law to hold for these inverses.

## Abstract

Within the field of multilinear algebra, inverses and generalized inverses of tensors based on the Einstein product have been investigated over the past few years. In this paper, we explore the singular value decomposition and full-rank decomposition of arbitrary-order tensors using {\it reshape} operation. Applying range and null space of tensors along with the reshape operation; we further study the Moore-Penrose inverse of tensors and their cancellation properties via the Einstein product. Then we discuss weighted Moore-Penrose inverses of arbitrary-order tensors using such product. Following a specific algebraic approach, a few characterizations and representations of these inverses are explored. In addition to this, we obtain a few necessary and sufficient conditions for the reverse-order law to hold for weighted Moore-Penrose inverses of arbitrary-order tensors.