All unmixed solutions of the algebraic Riccati equation using Pick matrices

In this short paper we study the existence of positive and negative semidefinite solutions of the algebraic Riccati equation corresponding to linear quadratic problems with an indefinite cost functional. An important role is played by certain two-variable polynomial matrices associated with the algebraic Riccati equation. We characterize al unmixed solutions in terms of the Pick matrices associated with these two-variable polynomial matrices. As a corollary it turns out that the signatures of the extremal solutions are determined by the signatures of particular Pick matrices.