# Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix

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.