# Second eigenvalue of a stochastic block matrix

Considering a stochastic block matrix in the form of, $$\textbf{P_{}} = \left( {\begin{array}{cc} \textbf{A_{}} & \textbf{B_{}}~; \ \textbf{B_{}} & \textbf{A_{}} \end{array} } \right) \label{P_RW}$$

I found out that the second largest eigenvalue $\lambda_{2}(P_{})$ of $P$ can be derived from, $\lambda_{2}(P_{}) = \textrm{max} \Big\{\lambda_{2}{(A_{}+B_{})}, \lambda_{max}(A_{}-B_{})\Big\}$

This can be easily done using the characteristic polynomial and properties of determinants. Since I'm involved in a paper which uses this property, I'd like to find this out in any reference (if available) to minimize the use of proof in the paper by refering it which I didn't succeed in finding out. Does anyone know any reference which has this proof done.?

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