## Calculating eigenvalues and eigenvectors

Consider the NxN matrix $G=\left( \begin{array}{cccc}
\alpha & & & 0 \\
\gamma & \alpha & & \\
& \ddots & \ddots & \\
0 & & \gamma & \alpha
\end{array} \right)$

I need to calculate its eigenvalues and eigenvectors. So far, I see that its characteristic polynomial is
$\Delta(t)=det(tI_{N}-G)=(t-\alpha)^{N}.$
The only root is $t=\alpha$, so the only eigenvalue is $\alpha$, and its corresponding eigenvector is $v=(v_{1},v_{2},...,v_{N-1},v_{N})^{T}=(0,0,...,0,v_{N})^{T}$, where $v_{N}$ is any non-zero real number.
I get that there is only one eigenvalue. Are my solutions right?