Considering a stochastic block matrix in the form of,
$$\textbf{$P_{}$} = \begin{pmatrix} \textbf{$A_{}$} & \textbf{$B_{}$} \\ \textbf{$B_{}$} & \textbf{$A_{}$} \end{pmatrix}$$
I found out that the second largest eigenvalue $\lambda_{2}(P_{})$ of $P$ can be derived from,
$$\lambda_{2}(P_{}) = \max \Big\{\lambda_{2}{(A_{}+B_{})}, \lambda_{\max}(A_{}-B_{})\Big\}$$
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?
