In my research I came across the following question.
Let $A$ be an integer non-negative matrix (every entry of $A$ is non-negative) and $x = (x_1,...,x_n)^T$ the probability Perron-Frobenius eigenvecor, i.e., $Ax = \lambda x$ and all $x_i \geq 0$. Denote by $H(x_1,...,x_n)$ the additive abelian group generated by $x_1,...,x_n$. Consider the set $\mathcal M(A)$ of all integer non-negative matrices $B$ such that $By = \lambda y$ where $y= (y_1,...,y_k)^T$ is the probability Perron-Frobenius eigenvecor. Suppose additionally that $H(x_1,...,x_n) =H(y_1,...,y_k)$. Is the set $\mathcal M(A)$ finite for every fixed $A$?
I'll be thankfull for any comments or suggestions.