ON NONNEGATIVE MATRICES WITH A FULLY CYCLIC PERIPHERAL SPECTRUM

Let $A$ be a square complex matrix. We denote by $\rho(A)$ the spectral radius of $A$. The set of eigenvalues of $A$ with modulus $\rho(A)$ is called the peripheral spectrum of $A$. The latter set is said to to be fully cyclic if whenever $\rho(A)\alpha x =Ax$, $x\neq 0$, $|a|= 1$, then $|x|(sgn \ x)^k$ is an eigenvector of $A$ corresponding to $\rho(A)\alpha^k$ for all integers $k$. In this paper we give some necessary conditions and a set of sufficient conditions for a nonnegative matrix to have a fully cyclic peripheral spectrum. Our conditions are given in terms of the reduced graph of a nonnegative matrix.