Norm of the stationary law for a doubly stochastic chain

A transition matrix is called doubly stochastic when all its entries are non-negative and each row and each column adds up to $1$. A Markov chain on the state space $\{1,2,\dots,100\}$ has a doubly stochastic transition matrix $P$, and $\pi$ denotes its stationary distribution. Compute the Euclidean norm $\lVert\pi\rVert$.

$1/10$. Because every column of $P$ sums to $1$, the uniform vector satisfies $\pi P=\pi$, so $\pi_i=\tfrac1{100}$ for each state. Its norm is $\sqrt{100\cdot 10^{-4}}=\tfrac1{10}$.