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 × n × n tensors on a space of n × 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 × n × n tensors.