Further results on Moore–Penrose inverses of tensors with application to tensor nearness problems

Abstract

The notion of the Moore–Penrose inverse of a matrix has been extended to tensor case with Einstein product, recently. In this paper, we continue this work and propose the full rank decomposition of tensors by introducing a new kind of tensor rank, which gives a novel representation of the Moore–Penrose inverse. Moreover, we also obtain several formulae related to the Moore–Penrose inverse of a tensor and its {i,j,k}-inverses. In addition, we consider the tensor nearness problem associated with a tensor equation, a generalization of matrix nearness problems that arise in many areas of applied matrix computations. It is shown that the unique solution to the tensor nearness problem can be represented by using the Moore–Penrose inverses of the given tensors.