We extend the vector cross-product to a mapping from Fs x Fm to Fn, where F= R or C. We derive the generalized Hurwitz matrix equation and an equivalence relation of the generalized Hurwitz-Radon matrices from the matrix representation of the mapping with respect to orthonormal bases. Then we use the basic matrix techniques to classify the Hurwitz-Radon matrices and the total invariants under this equivalence relation.
