Discrete J-spectral factorization of possibly singular polynomial matrices

We develop two state-space algorithms for discrete-time J-spectral factorization of possibly singular, possibly non-column-reduced, and with possible zeros at the origin para-hermitian matrices, by assignment of matrices in optimal LQ return difference equality. The column degrees of the spectral factor equal the column degrees of the given matrix and the dimension of a discrete-time algebraic Riccati system equals the sum of the column degrees. Necessary and sufficient condition for the least-order property of our state-space realizations is column reduction of the given matrix.