On the boundaries of the determinantal regions of ray pattern matrices

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 SA 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 SA to ray determinantal regions RA. 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 RA 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” (whose entries are all on the axes) of a complex matrix A, and show that 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 also shows that the study of the complex determinantal regions can be turned to the study of the ray determinantal regions.