Numerical range for weighted Moore-Penrose inverse of tensor

TL;DR

This paper introduces a new approach to compute the weighted Moore-Penrose inverse of tensors using weighted singular value decomposition and explores the numerical range properties of this inverse, extending matrix results.

Contribution

It develops the concept of weighted SVD for tensors and studies the numerical range of the weighted Moore-Penrose inverse, providing new theoretical insights and generalizations.

Findings

Defined weighted SVD for tensors using Einstein product

Established properties of the numerical range for the tensor inverse

Generalized existing matrix results to tensor setting

Abstract

This article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define the notions of weighted normal tensor for an even-order square tensor and weighted tensor norm. Finally, we apply these to study the theory of numerical range for the weighted Moore-Penrose inverse of an even-order square tensor and exploit its several properties. We also obtain a few new results in the matrix setting that generalizes some of the existing results as particular cases.