Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi = \text{diag}(\xi)$ to be a diagonal matrix with a principal left eigenvector $\xi$ of $P$ (i.e. $\xi^\top P = \xi^\top$) on the main diagonal and zeros elsewhere (i.e. the stationary distribution if the chain is ergodic). Denote by $\sigma(\cdot)$ the spectral radius of a matrix.
Prove the following spectral radius bound.
$\sigma(P^\top \Xi^2 P) \leq \max_i \xi_i^2$
I have tried the following approaches:
- $\| P^\top \Xi^2 P \|_2 \leq \| P^\top \|_2 \| \Xi^2 \|_2 \|P \|_2 = \| \Xi^2 \|_2 \|P \|_2^2 \leq (\sqrt2)^2 \max_i \xi_i^2 $ (not sharp enough).
- $\| P^\top \Xi^2 P \|_1 \leq \| P^\top \|_1 \| \Xi^2 \|_1 \|P \|_1 \leq n \max_i \xi_i^2 $ (even less sharp)
What about using the $\infty$-norm? That is $$ \|A\|_\infty = \sup_{x: \|x\|_\infty=1} \|Ax\|_\infty. $$ Take a vector $x$. Then $$ \|Px\|_\infty \le \max_{i}\left|\sum_j p_{ij} x_j\right| \le \max_{i}\sum_j p_{ij} (\max_k |x_k|) \le\|x\|_\infty. $$ Denote $z:=Px$. Then $$ \|P^T\Xi^2 z\|_\infty = \max_i \left|\sum_j p_{ji}\xi_j^2 z_j\right| \le\max_i \left|\sum_j p_{ji}\xi_j \right| \|\xi\|_\infty\|z\|_\infty. $$ Since $P^T\xi = \xi$, it follows $$ \sum_j p_{ji}\xi_j=\xi_i, $$ hence $\max_i \left|\sum_j p_{ji}\xi_j \right|=\|\xi\|_\infty$, which gives $$ \|P^T\Xi^2 Px\|_\infty = \|P^T\Xi^2 z\|_\infty\le \|\xi\|_\infty^2 \|z\|_\infty = \|\xi^2\|_\infty \|z\|_\infty, $$ which proves $$ \sigma(P^T\Xi^2 P) \le \|P^T\Xi^2 P\|_\infty \le \|\xi^2\|_\infty. $$