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## Matrix Limit

Suppose $M=\displaystyle\begin{pmatrix} p&q&r \\ r&p&q \\ q&r&p \end{pmatrix}$, where $p,q,r >0$ with $p+q+r=1$. Find $\lim\limits_{n \to \infty} M^n$ ( after proving that this limit exists )
Miklos Schweitzer 1950

Proof: Find the eigenvalues of the given matrix, see that they are differnt $1$ and two conjugate complex numbers which tend to 0 as $n \to \infty$, and therfore the matrix is diagonalizable and its powers converge.

To find its limit, find a recurence relation for the entries of the matrix power.