The purpose of this paper is to report some new results concerning the geometry of the set 𝒫0of all solutions of a quadratic matrix inequality (QMI) arising in stochastic realization theory and the subset 𝒫0of all solutions of the corresponding algebraic Riccati equation (ARE). The boundary, interior and extreme points of 𝒫 are studied. Some new results linking 𝒫0to 𝒫 are presented. A complete characterization of some subsets of 𝒫0and 𝒫 are given. Additional properties of the feedback matrix corresponding to any element in 𝒫0are obtained.