The determinantal regions of complex sign pattern matrices and ray pattern matrices

This paper studies the determinantal regions SA of complex sign pattern matrices and the determinantal regions RA of ray pattern matrices. We give an affirmative answer to a problem proposed in [C.A. Eschenbach, F.J. Hall, Z. Li, From real to complex sign pattern matrices, Bull. Australian Math. Soc. 57 (1998) 159-172] concerning the boundary points of the regions SA. We also prove that the region SA⧹{0} (and RA⧹{0}) is an open set in the complex plane C, except the cases that SA (or RA) is entirely contained in a line through the origin. We determine all the 21 possible regions SA of complex sign nonsingular matrices A and all the possible regions RA of ray nonsingular matrices A. We also study some further properties of SA (and RA) when A is not necessarily complex sign nonsingular (or ray nonsingular).