Geometric aspects of the Riccati difference equation in the nonsymmetric case (Q677798)

The nonsymmetric Riccati difference equation under consideration is \[ X(t+1)= [B_{21}(t)+ B_{22}(t) X(t)][B_{11}(t)+ B_{12}(t) X(t)]^{-1}, \] where \(X(t)\) is a complex matrix valued function of dimension \(m\times n\), on an interval \([t_0,t_1]\) of the integers, and \(B_{ij}(t)\) are similar matrix valued functions of suitable dimensions. This paper presents a study of the geometry of this equation. Two representations of the solutions are given in terms of some of them, assumed known. One formula allows to calculate all solutions in terms of \(k\) of them, where \(m\) and \(k\) are related. The other one identifies families of solutions which are projective superpositions of known solutions. The results generalize properties known for the symmetric Riccati equation (see, for example [\textit{C. V. de Souza}, Proceedings of the workshop on the Riccati equation in control, systems and signals, Como, Italy, 38-41 (1989)]). They are of importance in applications in the theory of control and differential games.