A note on numerical ranges of tensors

Abstract

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.

Keywords:

• Tensor
• Einstein product
• numerical range
• numerical radius
• Moore–Penrose inverse

MSC(2010):

• 15A69
• 15A09
• 47A12
• 47A30

Acknowledgements

The authors are thankful to Dr. Nachiketa Mishra for giving his precious time to a fruitful discussion. The authors also thank the anonymous referee for his/her comments that have significantly improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).