A circulant matrix is defined as: $C=\begin{bmatrix}c_{0}&c_{n-1}&\dots &c_{2}&c_{1}\\c_{1}&c_{0}&c_{n-1}&&c_{2}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{n-2}&&\ddots &\ddots &c_{n-1}\\c_{n-1}&c_{n-2}&\dots &c_{1}&c_{0}\\\end{bmatrix}$ where $C_{j, k}=c_{j-k \mod n}$. The $k$-th eigenvalue $\lambda_k$ and eigenvector $x_k$ satisfy $C\cdot x_k=\lambda_k x_k$, which can be expressed as $n$ equations: $\sum_{j=0}^{m-1}c_{m-j}x_j+\sum_{j=m}^{n-1}c_{n-j+m}x_j=\lambda_k x_m\quad m=0,1,\dots,n-1$ with $c_n=c_0$, where $x_m$ is the $m$-th component of the eigenvector $x_k$. By changing the […]