On the first degree Fejér–Riesz factorization and its applications to X + A⁎X−1A = Q

Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fejér–Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. This paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation X+A⁎X−1A=QX+A⁎X−1A=Q. In particular, the nonlinear equation can be recast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired.