A note on numerical ranges of tensors

Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the numerical range of a tensor via the k-mode product. However, the convexity of the numerical range via t...

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Bibliographic Details
Published in:Linear & multilinear algebra 2023-11, Vol.71 (16), p.2645-2669
Main Authors: Chandra Rout, Nirmal, Panigrahy, Krushnachandra, Mishra, Debasisha
Format: Article
Language:English
Subjects:
Algorithms
Convexity
Einstein product
Linear algebra
Mathematical analysis
Moore-Penrose inverse
numerical radius
numerical range
Tensor
Tensors
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Summary:Theory of numerical range and numerical radius for tensors is not studied much in the literature. Ke et al. [Linear Algebra Appl. 508 (2016), 100-132: MR3542984] introduced first the notion of the 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 via the Einstein product are introduced first. Using the notion of the numerical radius of a tensor, we provide some sufficient conditions for a tensor to be 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:0308-1087
1563-5139
DOI:10.1080/03081087.2022.2117771