## Perron–Frobenius theorem

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 eigenspace

–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 lecture. As pointed by Vaughn, this approach does not give precise information of the .

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Pengfei, on

May 23, 2015 at 10:05 pm, under Uncategorized. No Comments

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