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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 × 3 circulant matrix \begin{pmatrix}
1 & 2 & 3\\
3 & 1 & 2\\
2 & 3 & 1
\end{pmatrix}

For any n × n circulant matrix (n > 5), which of the following n-length vectors is always an eigenvector?

(a) A vector whose k-th element is k

(b) A vector whose k-th element is nk

(c) A vector whose k-th element is $\exp \left(j\dfrac{2π(n − 5)k}{ n}\right)$ where $j = \sqrt{−1}$

(d) A vector whose k-th element is $\sin h (2πk/n)$

(e) None of the above
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