TITLE: On positive matrices and their eigenvectors
QUESTION [7 upvotes]: Let $A$ be an $n\times n$ positive integer-valued matrix, that is every entry of $A$ is a a positive integer. Let $\lambda$ be the Perron-Frobenius eigenvalue and $x = (x_1,...,x_n)^T$ the corresponding positive probability eigenvector: $\sum_i x_i =1, \ x_i > 0$. Denote by $H(x)$ the additive subgroup of $\mathbb R$ whose generators are the coordinates of $x, \ 
H(x) = < x_1,...,x_n >$. 
Fix any integer $k \geq 1$ and consider the set of positive integer-valued matrices $\mathcal B_k$ formed by all $n\times n$ matrices $B$ satisfying the following conditions: $\lambda^k$ is the Perron-Frobenius eigenvalue for $B$, and if $By = \lambda^k y,\ \sum_i y_i = 1, y_i >0,$ then $H(y) = H(x)$.  
My questions are as follows. 
(1) Is there an algorithm describing all matrices from $\mathcal B_k$? 
(2) How can one find at least one matrix $B$ in the set $\mathcal B_k$ different from $PA^kP^{-1}$ where $P$ is a permutation matrix? 
Comments: (i) The case when $\lambda $ is an integer is not interesting, so that one can assume that  $\lambda$ is an algebraic number. (ii) I asked a similar question before but these ones seems formulated in more precise form. (iii) Of course, (2) is simpler than (1), and I actually need a  constructive answer to (2).
I'll be glad to see any comments, suggestions, references.

REPLY [2 votes]: This question is very closely related to problems in dimension groups (partially ordered abelian groups with various other properties), specifically stationary ones, for which there is a history going back to the early 80s [e.g., by me, Positive matrices and dimension groups affiliated to C*-algebras and topological Markov chains, J of Operator Theory 6 (1981) 55--74]. 
For example, if $A$ has determinant $\pm 1$, then $H(x)$ is an invariant of shift equivalence (related to conjugacy, but somewhat coarser). More generally (without the determinant condition), one considers $\cup_{n=0}^{\infty} H(x)\lambda^{-n}$, the direct limit group, an invariant of shift equivalence. In particular, examples size two examples (with irreducible characteristic polynomial) exist with the same $H$ values, but aren't shift equivalent. Of course, we can also arrange non-shift equivalence by making sure the spectra are different.
However, it is also true that if $\det A =\pm 1$ and $A$ and $B$ are primitive, and the characteristic polynomials of $A$ and $B$ are equal to each other and irreducible over the integers, then $H_A = H_B$ entails that $A$ is conjugate to $B$. This however, is rather special. Drop the irreducibility of the characteristic polynomial and the result fails, as the example above shows. Drop the determinant condition (keeping irreducibility of the char poly), and the union determines the shift equivalence class, almost the same as conjugacy. 
Likely what you are looking for is the connection between ideal class structure of orders in number fields and classification of the matrices.