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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?

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

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