# Reverse-order law for weighted Moore--Penrose inverse of tensors

## TL;DR

This paper explores properties and conditions under which the reverse-order law holds for the weighted Moore--Penrose inverse of tensors, extending understanding of tensor inverses via Einstein products.

## Contribution

It introduces new properties and sufficient conditions for the reverse-order law of the weighted Moore--Penrose inverse for tensors of arbitrary order.

## Key findings

- New properties of the weighted Moore--Penrose inverse for tensors
- Sufficient conditions for the reverse-order law in even-order square tensors
- Characterizations of the reverse-order law for tensors of any order

## Abstract

In this paper, we provide a few properties of the weighted Moore--Penrose inverse for an arbitrary order tensor via the Einstein product. We again obtain some new sufficient conditions for the reverse-order law of the weighted Moore--Penrose inverse for even-order square tensors. We then present several characterizations of the reverse-order law for tensors of arbitrary order.