The decomposability of the matrix with two determinantal regional components

The ray of a complex number a is either 0 or a/|a| depending on whether a is 0 or nonzero. The ray pattern of a complex matrix A, denoted by ray(A), is the matrix obtained by replacing each entry of A with its ray. The determinantal region of a square matrix A, denoted by RA, is the set of the determinants of all the complex matrices with the same ray pattern as A. A connected component of the set RA⧹{0} is called a determinantal regional component of A. The number of determinantal regional components of RA is denoted by nR(A). It was proved in Shao et al. [Jia-Yu Shao, Yue Liu, Ling-Zhi Ren, The inverse problems of the determinantal regions of ray pattern and complex sign pattern matrices, Linear Algebra Appl. 416 (2006) 835–843] that nR(A)≤2 for any complex square matrix A. When nR(A)=2, the two determinantal regional components are either two opposite open rays or two opposite open sectors with the angle no more than π. In this paper, we prove that any square matrix A with nR(A)=2 is partly decomposable if one of its determinantal regional components is an open sector with the angle less than π. As a main graph theoretical technique, we also discuss a property of strongly connected digraphs.