Consider the recurrence $g_0 = 1, g_1 = 3$ and for $n \geq 2,$ $$g_{n} = 3g_{n-1} - \frac{2}{3}g_{n-2}$$ I would like to find a closed-form solution for $g_n$, and I would imagine that it would be best to solve for this using a generating function method, but my friend recommended a matrix-based method.
I can see that
$$\left[\begin{array}{c} g_{n+1} \\ g_n\end{array}\right] = \left[\begin{array}{c} 3g_n - \frac{2}{3}g_{n-1} \\ g_n \end{array} \right] = \left[\begin{array}{cc} 3&\frac{-2}{3}\\ 1&0\end{array}\right] \left[\begin{array}{c} g_n \\ g_{n-1}\end{array}\right].$$
Hence, I can imagine that you can find $g_n$ using the statement $$\left[\begin{array}{c} g_n\\ g_{n-1} \end{array}\right] = \left[\begin{array}{cc} 3&\frac{-2}{3} \\ 1&0 \end{array}\right]^{n-1} \left[\begin{array}{c} g_1 \\ g_0\end{array}\right].$$
However, now that I have this, I cannot quite figure out how to calculate our given multiplication matrix by an arbitrary $n-1$ times. Is there possibly a way I can calculate this using matrix decomposition methods? If so, is there a particular method that you can show me would work reasonably well here?
To compute the Jordan canonical form of the involved $2\times 2$ matrix and exploit it to find $g_n$ is the same as finding the roots $\zeta_1,\zeta_2$ of the characteristic polynomial (of the sequence, that is the same as the characteristic polynomial of the matrix) $$ p(t) = t^2-3t+\frac{2}{3} $$ then state $$ g_n = C_1 \zeta_1^n + C_2 \zeta_2^n $$ where the absolute constants $C_1,C_2$ depend on $g_0,g_1$. In our case we have $$\zeta_1,\zeta_2 = \frac{9\pm\sqrt{57}}{6} $$ hence $$ g_n = \frac{3}{\sqrt{57}}\left[\left(\frac{9+\sqrt{57}}{6}\right)^{n+1}-\left(\frac{9-\sqrt{57}}{6}\right)^{n+1}\right].$$