We are learning about decoupling dynamical systems in linear algebra right now, and I am having some trouble with this problem. I don't understand what to do when we have a function $x(t+1)$, rather than the regular $x(t)$. Furthermore, reducing the matrix $A$ gives me complex eigenvalues. Would someone mind giving me direction (and/or a solution) for this problem? Thank you.
Find real-valued closed formulas for the trajectory $x(t+1)=Ax(t)$, where $A=\begin{bmatrix} 6 & -8\\ 8 & 6\\ \end{bmatrix}$ and $x⃗ (0)=\begin{bmatrix} 0\\ 1\\ \end{bmatrix}$
With $\theta=\arctan(4/3)$, we can write $$\begin{pmatrix}6&-8\\8&6\end{pmatrix}= 10\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}.$$ So $$\begin{pmatrix}6&-8\\8&6\end{pmatrix}^{n}= 10^{n}\begin{pmatrix}\cos n\theta&-\sin n\theta\\\sin n\theta&\cos n\theta\end{pmatrix}.$$ So $$x(n)=\begin{pmatrix}6&-8\\8&6\end{pmatrix}^{n} \begin{pmatrix}0\\1\end{pmatrix}= 10^{n}\begin{pmatrix}-\sin n\theta\\\cos n\theta\end{pmatrix}.$$