In [C.A. Eschenbach, F.J. Hall, Z. Li, From real to complex sign pattern matrices, Bull. Austral. Math. Soc. 57 (1998) 159-172], Eschenbach et al. proposed the problem concerning whether the boundaries of the complex determinantal regions S A are always on the axes in the complex plane. In [Jia-Yu Shao, Hai-Ying Shan, The determinantal regions of complex sign pattern matrices and ray pattern matrices, Linear Algebra Appl. 395 (2005) 211-228], an affirmative answer to this problem was obtained. In this paper, we generalize this result from complex determinantal regions S A to ray determinantal regions R A . Let T (A) be the set of the nonzero terms in the determinantal expansion of the matrix (A). Then we show that the boundary of the ray determinantal region R A is always a subset of the union of all those rays starting at the origin and passing through some one element of the set T (A).
We also define a so called "canonical form" A (whose entries are all on the axes) of a complex matrix A, and show that S A = R A for all complex square matrices A. Then the affirmative answer of the above problem will be a direct consequence of this and the result on the boundaries of the ray determinantal regions. This result S A = R A also shows that the study of the complex determinantal regions can be turned to the study of the ray determinantal regions.