Today I attended a lecture given by Vaughn Climenhaga. He presented a proof of the following version of Perron–Frobenius theorem:
Let be the set of probability vectors, be a stochastic matrix with positive entries. Then
–there is a positive probability fixed by
–the spectra for some
–for all , exponentially as .
Proof. (1) Let . Then , and
. So . Moreover, is positive and . Therefore there exists some point fixed by .
(2). Suppose on the contrary that there exists that is also fixed by . Then fixes every vector in the plane , in particular the points on . This contradicts (1).
(3). We use the norm . Note that . So . It suffices to show . If not, pick one ,say , and such that for any .
Consider the matrix , which is positive if is small enough. Then we have and hence . This contradicts the fact is an eigenvalue of .
(4). Let be the subset of vectors with zero mean: , and consider the decomposition . Note that and hence . For any , we have for some . Then .
Only light calculations are used in his lectue. As pointed by Vaughn, this approach does not give precise information of the .