How would you use eigenvalues and eigenvectors to compute a closed formula for the following sequence: $$\{x_0=1, x_1=2, x_n=5x_{n-1} + 14x_{n-2}, n \ge 0 \}$$
I have come up with the following formula: $$ \begin{bmatrix} 0 & 1 \\ 14 & 5 \end{bmatrix} * \begin{bmatrix} x_{n-2}\\ x_{n-1}\\ \end{bmatrix} = \begin{bmatrix} x_{n}\\ x_{n+1}\\ \end{bmatrix}$$
So to compute x_n I want to take the n'th power of the matrix A: $A=\begin{bmatrix} 0 & 1 \\ 14 & 5 \end{bmatrix}$ and multiply it with the "starting state" vector: $\begin{bmatrix} 1\\ 2\\ \end{bmatrix}$. Therefore I used the theory that $A^n=PD^nP^{-1}$, where $D$ is the diagonal matrix holding eigenvalues for $A$ along its diagonal and $P$ is the matrix holding the corresponding eigenvectors for the eigenvalues in $A$.
I computed the eigenvalues and vectors for $A$ and got that $\begin{bmatrix} 1\\ \frac{-1}{7} \end{bmatrix}$ is an eigenvector with eigenvalue $\lambda_1 = 7$. And $\begin{bmatrix} 1\\ \frac{1}{2} \end{bmatrix}$ is an eigenvector with eigenvalue $\lambda_2 = -2$.
So filling out the formula $A^n=PD^nP^{-1}$: $$ \begin{bmatrix} 0 & 1 \\ 14 & 5 \end{bmatrix}^n=\begin{bmatrix} 1 & 1 \\ \frac{-1}{7} & \frac{1}{2} \end{bmatrix} * \begin{bmatrix} 7^n & 0 \\ 0 & -2^n \end{bmatrix} * \frac{9}{14} \begin{bmatrix} \frac{1}{2} & -1 \\ \frac{1}{7} & 1 \end{bmatrix} $$ Which when multiplied gives: $$ \begin{bmatrix} 0 & 1 \\ 14 & 5 \end{bmatrix}^n=\frac{9}{14} * \begin{bmatrix} 7^n*\frac{1}{2}+(-2)^n *\frac{1}{7} & -9^n \\ -\frac{1}{7}*7^n*\frac{1}{2}+\frac{1}{2}*\frac{1}{7}*-2^n & \frac{1}{7}*-7^n+\frac{1}{2}*-2^n \end{bmatrix} $$
The problem is now that when I multiply the result onto the "starting vector" $\begin{bmatrix} 1\\ 2\\ \end{bmatrix}$ I get the this formula for $x_n$: $\frac{9}{14}((7^n+\frac{1}{2}+(-2)^n*\frac{1}{7})+(2*(-9)^n))$
Which doesn't give the correct answer when checking for $n=2$ for example: $\frac{9}{14}((7^2+\frac{1}{2}+(-2)^2*\frac{1}{7})+(2*(-9)^2)) = \frac{26721}{196}$. For $n=2$ I would expect $x_2=5*2 + 14*1=24$.
Can anyone help me figure out where I am doing it wrong?
Do you really need to use eigenvalues/eigenvectors? This is a linear difference equation for which you can easily identify a basis for the solution space. Looking at the roots of the characteristic polynomial $p(\lambda) = \lambda^2 -5 \lambda -14$, any solution is of the form $$ x_n = c_1 \cdot (-2)^n + c_2 \cdot 7^n $$
Now you can just compute $c_1, c_2$ from the initial condition $x_0 = 1$ and $x_1=2$.