Each factor contains no more than two nonzero entries in each row. Such a factorization is not unique, and this allows to obtain additional conditions. The perfect factorizations of the discrete Fourier transform matrices of third, fourth, fifth and sixth orders. The fast Fourier transform is based on factorization of the discrete Fourier transform matrix into the product of three matrices: the pre-summation matrix, the diagonal matrix and the post-summation matrix.