A note on numerical ranges of tensors

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.

arXiv comment20 pages

References in corpus (4)

• Further results on generalized inverses of tensors via the Einstein product
• Reverse order law for the Moore-Penrose inverses of tensors
• An extension of the Moore-Penrose Inverse of a Tensor via the Einstein product
• On reverse-order law of tensors and its application to additive results on Moore-Penrose inverse

Cited by in corpus (1)

• Numerical range for weighted Moore-Penrose inverse of tensor