Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (a"(3)) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (a"(3)) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.