Lets assume we have the following equation:

$AU=\lambda U \Rightarrow\left[ \begin{array}{c|c|c} 0 &A_{12}&A_{13}\\ \hline A_{21}& 0& A_{23}\\ \hline A_{31}&A_{32}&0 \end{array} \right]\left[ \begin{array}{c} u_1\\ \hline u_2\\ \hline u_3\\ \end{array} \right]=\lambda \left[ \begin{array}{c} u_1\\ \hline u_2\\ \hline u_3\\ \end{array} \right]$

where the left matrix, shown by $A$, is a non-negative irreducible block matrix and $\left[ \begin{array}{c} u_1\\ \hline u_2\\ \hline u_3\\ \end{array} \right]$ is a vector that is partitioned into subvectors $u_1$, $u_2$ and $u_3$. Clearly, this vector is the right eigenvector of $A$ with eigenvalue of $\lambda$.

I want to find a solution for $V$ and $\mu$ in the following equation versus $\lambda$ and $U$. In particular, I am looking at the case where $\lambda$ is the dominant eigenvalue of $A$ which, according to Perron Frobenius Theorem, is real and positive with positive eigenvector $U$.

$\left[ \begin{array}{c|c|c} 0 &A_{12}&A_{13}\\ \hline \mu A_{21}& 0& A_{23}\\ \hline \mu A_{31}&\mu A_{32}&0 \end{array} \right]\left[ \begin{array}{c} v_1\\ \hline v_2\\ \hline v_3\\ \end{array} \right]=\mu \left[ \begin{array}{c} v_1\\ \hline v_2\\ \hline v_3\\ \end{array} \right]$

This problem is related to my PhD research and any suggestion is highly appreciated.