$B$ is a non-negative irreducible block matrix as follows:

$$B= \left[ \begin{array}{c|c|c} 0 &B_{12}&B_{13}\\ \hline B_{21}& 0& B_{23}\\ \hline B_{31}& B_{32}&0 \end{array} \right].$$

Is there any relation between the maximum positive eigenvalue of $B$ and the maximum positive eigenvalue of the following matrix?

$$M= \left[ \begin{array}{c|c} B_{21}B_{12}& B_{21}B_{13}+B_{23}\\ \hline B_{31}B_{12}+B_{32}B_{21}B_{12}&B_{31}B_{13}+B_{32}B_{21}B_{13}+B_{32}B_{23} \end{array} \right]$$

censoringthe first block of states, in the language of Markov chains, i.e., $I-M$ is a Schur complement of $I-B$; in particular, if $B$ is stochastic then $M$ is, too, and the Perron eigenvector of $M$ is given by chopping off the last first block of that of $B$. $\endgroup$ – Federico Poloni Nov 20 '18 at 7:49