Third-order tensors as linear operators on a space of matrices

A recently proposed tensor tensor multiplication (M.E. Kilmer, C.D. Martin, L Perrone, A Third-Order Generalization of the Matrix SVD as a Product of Third-Order Tensors. Tech. Rep. TR-2008-4, Tufts University, October 2008) opens up new avenues to understanding the action of n x n x n tensors on a space of n x n matrices In particular it emphasizes the need to understand the space of objects upon which tensors act. This paper defines a free module and shows that every linear transformation on that module can be represented by tensor multiplication. In addition, it presents a generalization of ideas of eigenvalue and eigenvector to the space of n x n x n tensors. (C) 2010 Elsevier Inc. All rights reserved.