Is it true that for any square row-stochastic matrix one of the eigenvalues is $1$?

Question

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 row- or right-stochastic matrix (i.e. each row sums up to $1$) one of the eigenvalues is $1$? If so, how do we prove it?

Answer 1

That is a basic and important property of stochastic matrices. It's also non-obvious, unless you are aware of the Perron-Frobenius theorem.

Answer 2

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).