(1 point) Let M be a 3 x 3 matrix with eigenvalues $\lambda_1 = 0.8$, $\lambda_2 = -1$ $\lambda_3 = 1.8$ with corresponding eigenvectors
$\mathbf{v}_1 = \begin{bmatrix} 1 \ -1 \ 0 \end{bmatrix}$, $\mathbf{v}_2 = \begin{bmatrix} 0 \ 2 \ 2 \end{bmatrix}$, $\mathbf{v}_3 = \begin{bmatrix} 0 \ -1 \ 0 \end{bmatrix}$
Consider the difference equation
$\mathbf{x}_{k+1} = M \mathbf{x}_k$
with initial condition $\mathbf{x}_0 = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$.
Write the initial condition as a linear combination of the eigenvectors of M.
That is, write $\mathbf{x}_0 = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3 = \boxed{1} \mathbf{v}_1 + \boxed{3/2} \mathbf{v}_2 + \boxed{0} \mathbf{v}_3$
In general, $\mathbf{x}_k = \boxed{1} \left( \boxed{0.8} \right)^k \mathbf{v}_1 + \boxed{3/2} \left( \boxed{-1} \right)^k \mathbf{v}_2 + \boxed{0} \left( \boxed{1.8} \right)^k \mathbf{v}_3$
Specifically, $\mathbf{x}_5 = \begin{bmatrix} \boxed{0.3277} \ \boxed{-3.3277} \ \boxed{-3.0000} \end{bmatrix}$
For large k, $\mathbf{x}_k \approx \boxed{-3/2} \left( \boxed{1} \right)^k \begin{bmatrix} \boxed{0} \ \boxed{3} \ \boxed{3} \end{bmatrix}$
