Analogue of Perron numbers for non-negative matrices

Question

If $A$ is a square matrix with positive integer coefficients with a biggest real eigenvalues $\lambda>1$, then $\lambda$ is a Perron-number, which means that it is a real algebraic integer bigger than $1$, whose Galois conjugates $\mu$ satisfy $\lvert \mu\rvert<\lambda$. This follows from the Perron-Frobenius theorem.

The result is no longer true if $A$ is a square matrix with non-negative integer coefficients, as the simple example $\begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}$ shows: here $\lambda=\sqrt{2}$, which is not a Perron-number.

Here we know that the Galois-conjugates $\mu$ satisfy $\lvert \mu\rvert \le\lambda$ and that those where equality holds are simply $\lambda\theta$ where $\theta$ is a root of unity.

Is there then an analogue of the Perron-numbers for the real algebraic integers $\lambda>1$ that are biggest real eigenvalues of a square matrix with non-negative integer coefficients? I would be happy to have another name of such algebraic integers, and possibly literature about these numbers.

Answer 1

You are looking for "weak Perron numbers", which are numbers such that some positive integer power is a Perron number. In your example $\lambda=\sqrt{2}$, its square $\lambda^2=2$ is a Perron number, being an eigenvalue of a square matrix whose entries are all $2$.

For information, see Doug Lind's paper "The entropies of topological Markov shifts and a related class of algebraic integers", where Lind proves:

Theorem: $x \in \mathbb R$ is a weak Perron number if and only if $x$ is the dominant eigenvalue of some square matrix of non-negative integers.

For a later proof of this theorem see also William Thurston's paper "Entropy in dimension one".