# Proving that a matrix is nonnegative if its powers are nonnegative

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?

• Perhaps I am missing something but setting $k=1$ gives the converse surely? – copper.hat Feb 23 at 23:07
• Oh wow... you're right. Can't believe I missed that. Thanks! – Jordan Field Feb 23 at 23:11