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I have got the following transition matrix:

$$A = \begin{pmatrix} p & 1-p \\ 1-q & q \end{pmatrix}$$

How can one use the jordan normal form to get a closed-form to calculate such a values $$A^n_{i,j}$$ ?

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Did you try to compute the powers of a matrix in jordan normal form? What did you find? – Rasmus Jul 4 '12 at 12:19

You find $P$ and $B$ such that $B$ is in Jordan form and $A=PBP^{-1}$. Then you find a formula for $A^n$ involving $B^n$, and take the advice of Rasmus from the comments.

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I'll try. Thanks! – Heinrich Jul 4 '12 at 12:59

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