Spectral properties of one class of sign-symmertic matrices

TL;DR

This paper investigates the spectral properties of a class of sign-symmetric matrices, showing their similarity to nonnegative matrices and analyzing eigenvalue conditions under specific symmetry constraints.

Contribution

It establishes that J-sign-symmetric matrices are similar to nonnegative matrices and characterizes their eigenvalues, including conditions for complex eigenvalues with maximum modulus.

Findings

J-sign-symmetric matrices are similar to nonnegative matrices.

Existence of a second eigenvalue or multiple eigenvalues under certain conditions.

Conditions for complex eigenvalues with modulus equal to spectral radius.

Abstract

A $n\times n$ matrix $A$, which has a certain sign-symmetric structure ($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix is similar to a nonnegative matrix. The existence of the second in modulus positive eigenvalue $\lambda_2$ of a $J$--sign-symmetric matrix $A$, or an odd number $k$ of simple eigenvalues, which coincide with the $k$-th roots of $\rho(A)^k$, is proved under the additional condition that its second compound matrix is also $J$--sign-symmetric. The conditions when a $J$--sign-symmetric matrix with a $J$--sign-symmetric second compound matrix has complex eigenvalues, which are equal in modulus to $\rho(A)$, are given.