Extension of Moore–Penrose inverse of tensor via Einstein product

Abstract

ABSTRACT 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.