  In this paper, we provide a system theoretic treatment of a new class of
multilinear time-invariant (MLTI) systems in which the states, inputs and
outputs are tensors, and the system evolution is governed by multilinear
operators. The MLTI system representation is based on the Einstein product and
even-order paired tensors. There is a particular tensor unfolding which gives
rise to an isomorphism from this tensor space to the general linear group, i.e.
the group of invertible matrices. By leveraging this unfolding operation, one
can extend classical linear time-invariant (LTI) system notions including
stability, reachability and observability to MLTI systems. While the unfolding
based formulation is a powerful theoretical construct, the computational
advantages of MLTI systems can only be fully realized while working with the
tensor form, where hidden patterns/structures can be exploited for efficient
representations and computations. Along these lines, we establish new results
which enable one to express tensor unfolding based stability, reachability and
observability criteria in terms of more standard notions of tensor
ranks/decompositions. In addition, we develop a generalized CANDECOMP/PARAFAC
decomposition and tensor train decomposition based model reduction framework,
which can significantly reduce the number of MLTI system parameters. We
demonstrate our framework with numerical examples.
