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 $λ_2$ of a $J$--sign-symmetric matrix $A$, or an odd number $k$ of simple eigenvalues, which coincide with the $k$-th roots of $ρ(A)^k$, 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 $ρ(A)$, are given.

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