A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix.
\[\left(\begin{array}{lll}
1 & 2 & 3 \\
3 & 1 & 2 \\
2 & 3 & 1
\end{array}\right)\]
For any $n \times n$ circulant matrix $(n>5)$, which of the following $n$-length vectors is always an eigenvector?
- A vector whose $k$-th element is $k$
- A vector whose $k$-th element is $n^{k}$
- A vector whose $k$-th element is $\exp \left(j \frac{2 \pi(n-5) k}{n}\right)$ where $j=\sqrt{-1}$
- A vector whose $k$-th element is $\sinh \left(\frac{2 \pi k}{n}\right)$
- None of the above
