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 matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(s)</tex> 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 of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(s)</tex> are available. Furthermore, the calculation is extremely simple for the case, common in practice, that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi</tex> is given in the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(s) H^T (-s)</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> analytic in the open right half-plane. A technique is given for prefactoring an arbitrary <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(s)</tex> into this form.