Why isn't every eigenvalue of a stochastic matrix equal to 1? In this question, we see a proof that the largest eigenvalue of a stochastic matrix is equal to 1:
Proof that the largest eigenvalue of a stochastic matrix is 1
However, I think I've found a proof that every eigenvalue of a stochastic matrix is equal to 1.  Can you tell me where my proof is wrong?
Proof: Suppose ${\bf r}$ is an eigenvector of the column stochastic matrix $M$ (i.e. $M{\bf r} = \lambda {\bf r}$ for some $\lambda$), and assume without loss of generality that the entries of ${\bf r}$ sum to $1$.  Then $$M{\bf r} =
\begin{bmatrix}
M_{11}\\
M_{21}\\
\vdots\\
M_{n1}
\end{bmatrix} r_1 +
\begin{bmatrix}
M_{12}\\
M_{22}\\
\vdots\\
M_{n2}
\end{bmatrix} r_2
+ \dots +
\begin{bmatrix}
M_{1n}\\
M_{2n}\\
\vdots\\
M_{nn}
\end{bmatrix} r_n$$
Since $M$ is column stochastic, each column must sum to $1$, so the sum of the entries in $M {\bf r}$ is just $1 \cdot r_1 + 1 \cdot r_2 + \dots + 1 \cdot r_n = 1$.  Therefore, $\lambda$ must be $1$, and $M$ can only have one eigenvalue.
Thanks!

I think that you are assuming that the entries of $r$ are non-negative; in general, you cannot assume that the entries of $r$ add to 1, because they might add to zero. Take
$$
M=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}.
$$
Eigenvalues are $1$ and $0$. The eigenvector for $0$ is $\begin{bmatrix}1\\-1\end{bmatrix}$. 
