I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the answer in literature by myself.

My question is: does any of these statements hold?:

- all eigenvalues of a nonnegative (not necessarily irreducible) matrix are a $n$-th complex root of some real number for some $n \in \mathbb{N}$
- all leading eigenvalues of a nonnegative matrix are a $n$-th complex root of some real number for some $n \in \mathbb{N}$

If my knowledge is correct, the second statement should hold (because of any nonnegative matrix can be decomposed to a triangular block matrix with irreducible diagonal blocks). But I have no idea, if the first statement holds, and if so, why. Moreover, if the matrix is a nonnegative integer matrix, does it somehow simplify it's spectral properties?

I am interested in the problem because I am trying out to analyze a behavior of solutions of certain linear systems of recurrences, and any of these properties would significantly simplify the analysis.

Any ideas and references to literature would be extremely helpful. Thank you in advance.

Matrices, GTM 216, Springer-Verlag. – Denis Serre Jun 18 '12 at 8:20