Proof that the largest eigenvalue of a stochastic matrix is 11

The largest eigenvalue of a stochastic matrix (i.e. a matrix whose entries are positive and whose rows add up to 1) is 1.

Wikipedia marks this as a special case of the Perron-Frobenius theorem, but I wonder if there is a simpler (more direct) way to demonstrate this result.

Answer

Here’s a really elementary proof (which is a slight modification of Fanfan’s answer to a question of mine). As Calle shows, it is easy to see that the eigenvalue 1 is obtained. Now, suppose Ax=λx for some λ>1. Since the rows of A are nonnegative and sum to 1, each element of vector Ax is a convex combination of the components of x, which can be no greater than xmax, the largest component of x. On the other hand, at least one element of λx is greater than xmax, which proves that λ>1 is impossible.

Attribution
Source : Link , Question Author : koletenbert , Answer Author : Community

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