On the spectra of hypermatrix direct sum and Kronecker products constructions

We extend to hypermatrices definitions and theorem from matrix theory. Our main result is an elementary derivation of the spectral decomposition of hypermatrices generated by arbitrary combinations of Kronecker products and direct sums of cubic side length 2 hypermatrices. The method is based on a generalization of Parseval's identity. We use this general formulation of Parseval's identity to introduce hypermatrix Fourier transforms and discrete Fourier hypermatrices. We extend to hypermatrices a variant of the Gram–Schmidt orthogonalization process as well as Sylvester's classical Hadamard matrix construction. We conclude the paper with illustrations of spectral decompositions of adjacency hypermatrices of finite groups and a short proof of the hypermatrix formulation of the Rayleigh quotient inequality.