I was reading this paper "Fast Linear Iterations for Distributed Averaging" and on page 3 it reads that: $\lim_{t \to \infty} W^t$ exists (i.e., $W$ is semi-convergent) if and only if there is a nonsingular matrix $T$ such that $ W = T\begin{bmatrix} I_k & O\\ O & Z \end{bmatrix} T^{-1}$, where $\rho(Z)<1$, aka, spectral radius of $Z<1$.
I think the $\impliedby$ direction is easy to prove. Can anyone prove the $\implies$ direction? Thanks a lot.