Reverse-order law for the Moore–Penrose inverses of tensors is useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore–Penrose inverses of tensors. We then obtain a few necessary and sufficient conditions of the reverse-order law for the Moore–Penrose inverses of tensors via the Einstein product.

The notion of the Moore–Penrose inverse of an even-order tensor and the two-term reverse-order law for the Moore–Penrose inverse of even-order tensors via the Einstein product were introduced, very recently. In this article, the Moore–Penrose inverse of an arbitrary tensor is introduced first. Various new expressions of the Moore–Penrose inverse are also proposed. A new generalized inverse of a tensor called product Moore–Penrose inverse is then introduced by extending the Moore–Penrose inverse of an arbitrary tensor. A necessary and sufficient condition for the coincidence of the Moore–Penrose inverse and the product Moore–Penrose inverse of an arbitrary tensor is also provided. Prior to these, a set of new sufficient conditions for computing the Moore–Penrose inverse of the product of two tensors via the Einstein product is illustrated. Finally, some necessary and sufficient conditions are obtained for the three-term reverse-order law of tensors via the same product.

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.