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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.

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