Is this the right way to do this?

Define the system of equations;

$$x_{n+1} = x_n - y_n$$

$$y_{n+1} = 2x_n + 4y_n$$

so if we take $$A = \begin{pmatrix}
1 & -1\\
2 & 4
\end{pmatrix}$$

and define

$$\begin{pmatrix}
x_{n+1} \\
y_{n+1} \\
\end{pmatrix} = \begin{pmatrix}
1 & -1 \\
2 & 4 \\
\end{pmatrix}\begin{pmatrix}
x_n \\
y_n \\
\end{pmatrix}$$

Then given that the eigenvalues and a corresponding eigenvector to the eigenvalue can be defined as

$$\lambda_1 = 3$$

$$\mu_1 = \begin{pmatrix}
1 \\
-2 \\
\end{pmatrix}$$

$$\lambda_2 = 2$$

$$\mu_2 = \begin{pmatrix}
1 \\
-1
\end{pmatrix}$$

(I already checked this)

Is it then true that the general solution can be defined as

$$x_n = 3^nc_1 + 2^nc_2$$

$$y_n = -2*3^nc_1 - 2^nc_2$$

Reinout