This paper considers the set (℘) of all solutions of a quadratic matrix inequality (QMI) that arises in the discrete-time stochastic realization theory and the subset (℘0) of all solutions of the corresponding algebraic Ricatti equation (ARE). A complete characterization of some subsets of ℘ and ℘0is given. The boundary, interior and extreme points of ℘ are identified. It is shown that elements of ℘0are not only boundary points of the convex compact set ℘, but are extreme points as well.