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I have a stochastic matrix $A \in R^{n \times n}$ whose sum of the entries in each row is 1. When I found out the eigenvalues and eigenvectors for this stochastic matrix, it always happens that one of the eigenvalues is $1$.

Is it true that for any square stochastic matrix, one of the eigenvalues is $1$? If so, how do we prove it?

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Clarification: are we speaking of a "left stochastic matrix" (each row sums 1) and left eigenvectors/eigenvalues?… – leonbloy Aug 27 '11 at 18:00
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That an a basic and important property of stochastic matrixes. It's also non-obvious, unless you are aware of the Perron-Frobenius theorem.

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What is non-obvious? The property the OP asks about (that 1 is an eigenvalue) IS obvious once one notices that AU=U for U the column vector of ones... :-) – Did Aug 27 '11 at 17:00
I'm assuming, when he says that the rows sum one, that we are dealing with a "right stochastic matrix", and hence we are speaking of "left eigenvalues" If we have a "left stochastic matrix" that additionally has the property of its rows summing one, then it's a double strochastic matrix ( a very particular case) and then you're right, it's obvious. – leonbloy Aug 27 '11 at 17:48
No particular case. Since AU=U, 1 is an eigenvalue of A (for example in the sense that ker(A-I) is not {0}), hence det(I-A)=0, hence det(I-A^T)=0 (because det(M)=det(M^T) for every matrix M), hence 1 is an eigenvalue of A^T, hence there exists a nonzero vector V such that A^TV=V, hence V^TA=V^T. – Did Aug 27 '11 at 18:01

I was going to make this a comment, but I might as well make it an answer. The column vector with every entry 1 is an eigenvector with eigenvalue $1$ for your matrix. It is not necessarily true that the eigenvalue $1$ occurs with algebraic multiplicity $1$ as an eigenvalue for your matrix $A$. By the Frobenius Perron-Theorem, that is the case if the entries of $A$ (or even some power of $A$) are all positive. What is true is that $x-1$ is not a repeated factor of the minimum polynomial of $A$ (using the Frobenius Perron theorem on each block).

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