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Eigen vectors of $$\begin{bmatrix}1 & \cos \theta \\\cos \theta &1 \end{bmatrix}$$ are
$$\begin{bmatrix}a^{n} & 1 \\ 0 & a^{n} \end{bmatrix} $$
$$\begin{bmatrix}a^{n} &n \\ 0 & a^{n} \end{bmatrix} $$
$$\begin{bmatrix}a^{n} &n.a^{n-1} \\ 0 & a^{n} \end{bmatrix}$$
$$\begin{bmatrix}a^{n} &n.a^{n-1} \\ -n & a^{n} \end{bmatrix}$$
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