Let M be a 2 x 2 matrix with eigenvalues $\lambda_1 = -2$, $\lambda_2 = 1.5$ with corresponding eigenvectors
$\mathbf{v}_1 = \begin{bmatrix} -2\\2 \end{bmatrix}$, $\mathbf{v}_2 = \begin{bmatrix} 0\\-2 \end{bmatrix}$.
Consider the difference equation
$\mathbf{x}_{k+1} = M\mathbf{x}_k$
with initial condition $\mathbf{x}_0 = \begin{bmatrix} 7\\7 \end{bmatrix}$.
Write the initial condition as a linear combination of the eigenvectors of M.
That is, write $\mathbf{x}_0 = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 = \Box \mathbf{v}_1 + \Box \mathbf{v}_2$
In general, $\mathbf{x}_k = \Box \left(\Box\right)^k \mathbf{v}_1 + \Box \left(\Box\right)^k \mathbf{v}_2$
Specifically, $\mathbf{x}_5 = \begin{bmatrix} \Box\\\Box \end{bmatrix}$
For large k, $\mathbf{x}_k \approx \Box \left(\Box\right)^k \begin{bmatrix} \Box\\\Box \end{bmatrix}$
