Perturbation theory for Moore–Penrose inverse of tensor via Einstein product

Abstract

The spectral norm of an even-order tensor is defined and investigated. An equivalence between the spectral norm of tensors and matrices is given. Using derived representations of some tensor expressions involving the Moore–Penrose inverse, we investigate the perturbation theory for the Moore–Penrose inverse of tensor via Einstein product. The classical results derived by Stewart (SIAM Rev 19:634–662, 1977) and Wedin (BIT 13:217–232, 1973) for the matrix case are extended to even-order tensors. An implementation in the Matlab programming language is developed and used in deriving appropriate numerical examples.