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 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.

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