Q-matrices and spherical geometry

A real n × n matrix M is a Q-matrix if the linear complementarity problem w ? Mz=q, w ? 0, z ? 0, w^tz=0 has a solution for all real n-vectors q. M is nondegenerate if all its principal minors are nonzero. Spherical geometry is applied to the problem of characterizing nondegenerate Q-matrices. The stability of 3 × 3 nondegenerate Q-matrices and a generalization of the partitioning property of P-matrices are rather easily proved using spherical geometry. It is also proved that the set of 4 × 4 nondegenerate Q-matrices is not open.