A note on numerical ranges of tensors

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 v...

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Bibliographic Details
Published in:arXiv.org 2021-08
Main Authors: Rout, Nirmal Chandra, Panigrahy, Krushnachandra, Mishra, Debasisha
Format: Article
Language:English
Subjects:
Algorithms
Convexity
Linear algebra
Mathematical analysis
Tensors
Online Access:Get full text
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Summary: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.
ISSN:2331-8422