I have conjectured that the eigenvalues of the following $N\times N$ matrix are all distinct:
$$\mathbf{A}=\left[\begin{array}{ccccccc} \sin\left(\frac{2\pi 0}{N} \right)&-\frac{1}{2}&&&&&\frac{1}{2}\\ \frac{1}{2}&\sin\left(\frac{2\pi 1}{N} \right)&-\frac{1}{2}&&&&\\ &\frac{1}{2}&\sin\left(\frac{2\pi 2}{N} \right)&\ddots&&&\\ &&\ddots&\ddots&\ddots&&\\ &&&\ddots&\ddots&-\frac{1}{2}&\\ &&&&\frac{1}{2}&\sin\left(\frac{2\pi (N-2)}{N} \right)&-\frac{1}{2}\\ -\frac{1}{2}&&&&&\frac{1}{2}&\sin\left(\frac{2\pi (N-1)}{N} \right) \end{array}\right].$$ Can anyone help me in providing a proof regarding this fact?
The omitted components of $A$ are all zero.