The determinantal regions of complex sign pattern matrices and ray pattern matrices
Let \(A\) be a complex square matrix and let \(S_A\) (resp. \(R_A\)) be the set of determinants of the matrices in the complex sign (resp. ray) pattern class of \(A\). \textit{C. A. Eschenbach}, \textit{F. J. Hall} and \textit{F. Z. Li} [Bull. Aust. Math. Soc. 57, No. 1, 159--172 (1998; Zbl 0951.15021)] mentioned several open problems concerning the determinantal region \(S_A\). The authors show that the boundary points of the region \(S_A\) lie always on some axis of the complex plane. Moreover, if \(S_A\) (resp. \(R_A\)) is not contained in any line through the origin, then all nonzero boundary points of \(S_A\) (resp. \(R_A\)) are not in \(S_A\) (resp. \(R_A\)). Some consequences are discussed. The authors also find bounds for the number of determinantal regions. In the case of a nonsingular matrix \(A\), all the \(21\) possible determinantal regions \(S_A\) are determined and listed. Some research problems concerning the regions \(R_A\) are proposed.
scientific article; zbMATH DE number 2137797