My question is actually about the derivation of eigenvalues and eigenvectors of the circulant matrix. I am not good at doing that in a straight way, i.e. calculating them with a general form of circulant matrix. Instead, after reading alot, I tried calculating them of a permutation matrix $P$ which is a special case.
Let $$P= \left[ \begin{matrix} 0 & 0 & 0 & \cdots & 1 \\ 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \cdots & 1 \end{matrix} \right], $$ the eigenvalues of $P$ is simply given by $$ \left|P-\lambda I \right| = \left| \begin{matrix} -\lambda & 0 & 0 & \cdots & 0 & 1 \\ 1 & -\lambda & 0 & \cdots & 0 & 0 \\ 0 & 1 & -\lambda & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & 0 & 0 \\ 0 & 0 & 0 & \ddots & -\lambda & 0 \\ 0 & 0 & 0 & \cdots & 1 & -\lambda \end{matrix} \right| = 1-\lambda^n. $$ Then we know that the eigenvalues are simply n roots of unit of 1 which could be $\varepsilon_k=exp(\frac{2k\pi i}{n}), k=0, 1, \cdots, n-1$, i.e. $\lambda_k = \varepsilon^k, \varepsilon = exp(\frac{2\pi i} {n}$).
Later, I tried getting derivation of the eigenvectors $\mathbf{u}_k=[u_{k1}, u_{k2}, \cdots, u_{kn}]^T$ with $[P-\lambda_k I]\mathbf{u}_k=0$, i.e. $$ [P-\lambda_k I]\mathbf{u}_k = \left[ \begin{matrix} -\lambda_k & 0 & 0 & \cdots & 0 & 1 \\ 1 & -\lambda_k & 0 & \cdots & 0 & 0 \\ 0 & 1 & -\lambda_k & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & 0 & 0 \\ 0 & 0 & 0 & \ddots & -\lambda_k & 0 \\ 0 & 0 & 0 & \cdots & 1 & -\lambda_k \end{matrix} \right] \left[ \begin{matrix} u_{k1} \\ u_{k2} \\ u_{k3} \\ \vdots \\ u_{k(n-1)} \\ u_{kn} \end{matrix} \right]. $$ This equation gives a set of formulas of $\lambda_k$ $$ \begin{matrix} -\lambda_k u_{k1} + u_{kn} \\ u_{k1} - \lambda_k u_{k2} \\ u_{k2} - \lambda_k u_{k3} \\ \vdots \\ u_{k(n-1)} - \lambda_k u_{kn} \end{matrix} = 0. $$ By some of reference articles, one way is to assume that $u_{k1}=1$ which hence gives the first eigenvector $\mathbf{u}_0 = [1,1,\cdots,1]^T$. However, I failed to get other eigenvectors in the form of $[1, \varepsilon^{k1}, \varepsilon^{k2}, \cdots, \varepsilon^{k(n-1)}]^T$.