A solution X of a discrete-time algebraic Riccati equation is called unmixed if the corresponding closed-loop matrix Φ( X) has the property that the common roots of det( sI− Φ( X)) and det(I−sΦ(X) ∗) (if any) are on the unit circle. A necessary and sufficient condition is given for existence and uniqueness of an unmixed solution such that the eigenvalues of Φ( X) lie in a prescribed subset of C .