Ray patterns of matrices and nonsingularity

A complex matrix A is ray-nonsingular if det(X A) ≠ 0 for every matrix X with positive entries. A sufficient condition for ray nonsingularity is that the origin is not in the relative interior of the convex hull of the signed transversal products of A. The concept of an isolated set of transversals is defined and used to obtain a necessary condition for A to be ray-nonsingular. Some fundamental similarities as well as differences between ray nonsingularity and sign nonsingularity are illustrated.