Consider the NxN matrix G=\left( \begin{array}{cccc}<br />
\alpha & & & 0 \\<br />
\gamma & \alpha & & \\<br />
 & \ddots & \ddots & \\<br />
0 & &  \gamma & \alpha<br />
\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?