Geometry of the Algebraic Riccati Equation, Part I

We synthesize and generalize the two principal methods for classifying the set of real symmetric solutions of the algebraic Riccati equation (ARE) to obtain a result which combines the advantages of both existing methods. The geometric approach we take also clarifies the roles of controllability and observability in the theory of the ARE. In addition, we discuss the properties of a special subset of the solution set of the ARE, the unmixed solutions, and find that they exhibit many of the same properties as the extremal solutions, K^ + and K^ -.