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 ≤ 13. ∗This paper was written while Professor Lee was visiting the University of Wyoming and was supported by a 1996 Post-doctoral Fellowship from KOSEF. †Work supported by a National Science and Engineering Research Council of Canada grant.