The spectral factorization of para-Hermitian matrices is often required in problems of filtering theory, network synthesis, and control systems design. An algebraic factorization technique is developed. The factorization of the matrixPhi(s)is shown to be directly determined by the unique solution of a system of linear matrix equations. The labor of the method is reduced when some forms of prefactorization ofPhi(s)are available. Furthermore, the calculation is extremely simple for the case, common in practice, thatPhiis given in the formH(s) H^T (-s)withHanalytic in the open right half-plane. A technique is given for prefactoring an arbitraryPhi(s)into this form.