Some Perturbation Theorems for Q-Matrices

Given a real $n \times n$ matrix M and vector q, the linear complementarily problem is to find vectors w and z such that $w - Mz = q$, $w\geqq 0$, $z\geqq 0$, $w^t z = 0$. M is nondegenerate if all its principal minors are nonzero, and is a Q-matrix if the above problem has a solution for all $q \in E^n $. The principal result is that the set of Q-matrices is not open, but the set of nondegenerate Q-matrices is open. This is the foundation of a homotopy-like perturbation theorem. Finally, some related topological theorems are stated.