A note on numerical ranges of tensors

TL;DR

This paper introduces the concepts of numerical range and numerical radius for even-order square tensors using the Einstein product, proves their convexity, and explores properties related to tensor inverses, providing algorithms for visualization.

Contribution

It defines numerical range and radius for tensors via Einstein product, proves convexity, and discusses properties of tensor inverses, filling gaps in tensor numerical range theory.

Findings

Numerical range for tensors is convex.

Conditions for tensor unitarity using numerical radius.

Algorithm developed for plotting tensor numerical range.

Abstract

Theory of numerical range and numerical radius for tensors is not studied much in the literature. In 2016, Ke {\it et al.} [Linear Algebra Appl., 508 (2016) 100-132] introduced first the notion of numerical range of a tensor via the $k$-mode product. However, the convexity of the numerical range via the $k$-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors using inner product via the Einstein product are introduced first. We provide some sufficient conditions using numerical radius for a tensor to being unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore--Penrose inverse of a tensor are discussed.