On reverse-order law of tensors and its application to additive results on Moore–Penrose inverse

Abstract

The equality $(\mc{A}\n\mc{B})^\dagger = \mc{B}^\dagger \n \mc{A}^\dagger$ for any two complex tensors $\mc{A}$ and $\mc{B}$ of arbitrary order, is called as the {\it reverse-order law} for the Moore-Penrose inverse of arbitrary order tensors via the Einstein product. Panigrahy {\it et al.} [Linear Multilinear Algebra; 68 (2020), 246-264.] obtained several necessary and sufficient conditions to hold the reverse-order law for the Moore-Penrose inverse of even-order tensors via the Einstein product, very recently. This notion is revisited here among other results. In this context, we present several new characterizations of the reverse-order law of arbitrary order tensors via the same product. More importantly, we illustrate a result on the Moore-Penrose inverse of a sum of two tensors as an application of the reverse-order law which leaves an open problem. We recall the definition of the Frobenius norm and the spectral norm to illustrate a result for finding the additive perturbation bounds of the Moore-Penrose inverse under the Frobenius norm. We conclude our paper with the introduction of the notion of sub-proper splitting for tensors which may help to find an iterative solution of a tensor multilinear system.