I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if $P$ is doubly stochastic, then $P^k$ is, but I'm a little stumped on the converse.
What I have learned so far is that since $P^k$ has $\lambda_{PF} = 1$ (Perron-Frobenius eigenvalue) with $v_{PF} = 1$ (positive eigenvector of all 1's), that the same is also true for P (through analysis of the Jordan normal form). In short, I can now prove that the rows and columns of P add up to 1.
However, I believe I must also prove that $P$ is a nonnegative matrix if $P^k$ is. Can someone help me?