Extremal properties of ray-nonsingular matrices

A ray-nonsingular matrix is a square complex matrix, A, such that each complex matrix whose entries have the same arguments as the corresponding entries of A, is nonsingular. Extremal properties of ray-nonsingular matrices are studied in this paper. Combinatorial and probabilistic arguments are used to prove that if the order of a ray-nonsingular matrix is at least 6, then it must contain a zero entry, and that if each of its rows and columns have an equal number, k, of nonzeros, then k less than or equal to 13. (C) 2000 Elsevier Science B.V. All rights reserved.