On the properties of a new tensor product of matrices

Previously, the author introduced a new tensor product of matrices according to which the matrix of the discrete Walsh–Paley transform can be represented as a power of the second-order discrete Walsh transform matrix $H$ with respect to this product. This power is an analogue of the representation of the Sylvester–Hadamard matrix in the form of a Kronecker power of $H$. The properties of the new tensor product of matrices are examined and compared with those of the Kronecker product. An algebraic structure with the matrix $H$ used as a generator element and with these two tensor products of matrices is constructed and analyzed. It is shown that the new tensor product operation proposed can be treated as a convenient mathematical language for describing the foundations of discrete Fourier analysis.