The matrix A below has eigenvalues 1 , \frac{2}{3} , \text{and} \frac{1}{3} , with corresponding eigenvectors v_{1} , v_{2} , \text{and} v_{3} :
A = \frac{1}{9}\left [ \begin{matrix} 7 & -2 & 0 \\ -2 & 6 & 2 \\ 0 & 2 & 5 \end{matrix} \right ] , v_{1} = \left [ \begin{matrix} -2 \\ 2 \\ 1 \end{matrix} \right ] , v_{2} = \left [ \begin{matrix} 2 \\ 1 \\ 2 \end{matrix} \right ] , v_{3} = \left [ \begin{matrix} 1 \\ 2 \\ -2 \end{matrix} \right ]
Find the general solution of the equation x_{k+1} = Ax_{k} \text{if} x_{0} = \left [ \begin{matrix} 1 \\ 11 \\ -2 \end{matrix} \right ] .