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

TL;DR

This paper introduces a new class of sign-symmetric matrices called J--sign-symmetric, studies their spectral properties, especially eigenvalues, and explores approximation methods with strictly J--sign-symmetric matrices.

Contribution

The paper defines J--sign-symmetric matrices, analyzes their spectra, and provides conditions for eigenvalues and approximation by strictly J--sign-symmetric matrices.

Findings

Conditions for complex eigenvalues on the spectral circle.

Existence of two positive simple eigenvalues under certain conditions.

Feasibility of approximating J--sign-symmetric matrices with strictly J--sign-symmetric matrices.

Abstract

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.