  This paper investigates the properties of the solutions of the generalised
discrete algebraic Riccati equation arising from the solution of the classic
infinite-horizon linear quadratic control problem. In particular, a geometric
analysis is used to study the relationship existing between the solutions of
the generalised Riccati equation and the output-nulling subspaces of the
underlying system and the corresponding reachability subspaces. This analysis
reveals the presence of a subspace that plays an important role in the solution
of the related optimal control problem, which is reflected in the generalised
eigenstructure of the corresponding extended symplectic pencil. In establishing
themain results of this paper, several ancillay problems on the discrete
Lyapunov equation and spectral factorisation are also addressed and solved.
