Compound Product of Hypermatrices

The semi-tensor product (STP) of matrices is generalized to multidimensional arrays, called the compound product of hypermatrices. The product is first defined for three-dimensional hypermatrices with compatible orders and then extended to general cases. Three different types of hyperdeterminants are introduced and certain properties are revealed. The Lie groups and Lie algebras corresponding to the hypermatrix products are constructed. Finally, these results are applied to dynamical systems.