# What is the eigenvalue of stochastic matrix?

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? en.wikipedia.org/wiki/… – leonbloy Aug 27 '11 at 18:00

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).