Symmetric nonsquare factorization of selfadjoint rational matrix functions and algebraic Riccati inequalities

In this paper we shall present a parametrization of all symmetric, possibly nonsquare minimal factorizations of a positive semidefinite rational matrix function. It turns out that a pole-pair of such a nonsquare factor is the same as a pole pair for a specific square factor. The location of the zeros is then determined by a solution to a certain algebraic Riccati inequality. We shall also consider the case where the function we wish to factorize in a symmetric way has only constant signature. A connection with Bezoutians is given as well.