<s>[INST] Does this matrix sequence always converge? Suppose $a_0, a_1, ... , a_{n-1}$ are real numbers from $(0; 1)$, such that $\sum_{k=0}^{n-1} a_k=1$. Suppose $A = (c_{ij})$ is a $n \times n$ matrix with entries $c_{ij} = a_{(i-j)\%n}$, where $\%$ is modulo operation. Is it always true that $\lim_{m \to \infty} A^m = \frac{1}{n} \begin{pmatrix} 1 & 1 & \cdots & 1 \\1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{pmatrix}$?
This statement is true for $n = 2$: 
Suppose $A = \begin{pmatrix} a_0 & {1 - a_0} \\ {1 - a_0} & a_0 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 1 & 1\end{pmatrix}^{-1}\begin{pmatrix} 1 & 0 \\ 0 & {1-2a_0} \end{pmatrix}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$. Because $a_0$ is in $(0; 1)$, $1-2a_0$ is in $(-1;1)$. Thus $$\lim_{m \to \infty} A^m = \begin{pmatrix} 1 & -1 \\ 1 & 1\end{pmatrix}^{-1}\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$
However, I do not know how to prove this statement for arbitrary $n$.
Any help will be appreciated. 
 [/INST] This follows directly from the Perron-Frobenius theorem.  
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