# Linear algebra calculation using Markov process

I have a question about computing the following linear algebraic operation

Suppose

$u = (1, 0, 0)$

$v = \begin{pmatrix} 1\\ 0\\ 0\end{pmatrix}$

$A = \begin{pmatrix} 0.5 &0.2 &0.3\\0.2 &0.3 &0.5\\ 0.3 &0.5 &0.2\end{pmatrix}$

and

$g(n) = u\cdot A^n\cdot v$

I want to calculate

$g(20)/g(19)$ constructing a Markov process and using Metropolis algorithm.

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I know that if you can find a basis $x,y,z$ for ${\bf R}^3$ consisting of eigenvectors of $A$ with eigenvalues $a,b,c$, respectively, then $$A^nv=d_1a^nx+d_2b^ny+d_3c^nz$$ where you get the numbers $d_1,d_2,d_3$ from $v=d_1x+d_2y+d_3z$. Once you have $A^nv$ there is no difficulty getting $g(n)=uA^nv$, and then getting $g(20)/g(19)$.
Of course, since $A$ is stochastic and regular, $A^n$ converges, and I'd expect $g(20)/g(19)$ to be pretty close to $1$.