On spectrum and approximations of one class of sign-symmetric matrices

A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J–sign-symmetric. The spectrum of a J–sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also J–sign-symmetric and irreducible. The conditions, when such matrices have complex eigenvalues on the largest spectral circle, are given. The existence of two positive simple eigenvalues ${\lambda }_1>{\lambda }_2>0$ of a J–sign-symmetric irreducible matrix A is proved under some additional conditions. The question, when the approximation of a J–sign-symmetric matrix with a J–sign-symmetric second compound matrix by strictly J–sign-symmetric matrices with strictly J–sign-symmetric compound matrices is possible, is also studied in this paper.