Spectral properties of one class of sign-symmertic matrices

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.