linear algebra - Non-negative matrices with prescribed Perron-Frobenius eigenvectors

Without the assumption of irreducibility of $B$, there seems to be the following counterexample, even if we restrict to $x_i,y_j>0$. Let $A=E_{12}+E_{21}+E_{22}$, a 2 by 2 matrix. Then $lambda$ is the golden ratio, $x_1=lambda^{-2}$, $x_2=lambda^{-1}$, and so the group $H$ generated by $x_1$, $x_2$ is the ring of integers ${mathcal O}$ of the real quadratic field ${Bbb Q}(lambda)$. The ring ${mathcal O}$ is dense in the real line, so for any $n$ there are $n$-tuples $(a_1,...,a_n)in (0,1)cap {mathcal O}$ such that $a_1+...+a_n=1$. Then the vectors $(a_1x_1,a_1x_2,a_2x_1,a_2x_2,...,a_nx_1,a_nx_2)^T$ work as vectors $y$ for $B=Aoplus...oplus A$ ($n$ times). (As I understand, you allow $k$ to vary. Otherwise there are finitely many $B$ without any assumption on $H$ since the norm of $B$ is $lambda$, where I use the norm $||z||:=sum y_i|z_i|$ on row vectors. Indeed, the set of such $B$ is both discrete and compact.)