Given a stochastic matrix of $\mathbb{R}^{n\times n}$, say $\mathcal{A}(t)$. Let $$ S(t)=\begin{bmatrix} (1-r)I& rI\\-rI&rI+\mathcal{A}(t) \end{bmatrix} $$ where $r\in(0,1)$, $I$ is the identity matrix of $\mathbb{R}^{n\times n}$.
Estimate the spectral radius of $S(t)$. (About $\mathcal{A}(t)$, we only known that it is stochastic)
What I have tried is the following:
Let $S_{11}(t)=I-rI,S_{12}(t)=rI=-S_{21}(t),S_{22}(t)=rI+\mathcal{A}(t)$. Define $s(t)=[\|S_{ij}(t)\|_2]$. I use the spectral radius of $s(t)$, which is $1+\sqrt{2}r$ as the estimate of spectral radius of $S(t)$. But in the estimation, the sign of $S_{21}(t)$ have been ignored, which causes the conservativeness of this estimation.
So I wonder if there is a better estimation for the spectral radius.