I am trying to find out the spectrum (the collection of eigenvalues) with their multiplicities of the cycle graph $C_n$. Assuming that $X=\pmatrix{x_1\\x_2\\\vdots\\x_n}$ is the eigenvector, by considering $AX=\lambda X$, where $A$ is the adjacency matrix, I get the system of (non linear) equations:
$x_n+x_2=\lambda x_1\\x_1+x_3=\lambda x_2\\\cdots\\x_{n-1}+x_1=\lambda x_n$.
There seem to be no obvious way to solve this except summing which yields $\lambda=2$, provided $\sum x_i\ne 0$. However I already knew that $2$ is an eigenvalue of multiplicity $1$ as $C_n$ is a connected $2$-regular graph.
How do I go about finding the other eigenvalues?
Also in my book the solution provided magically says let $x_i=\epsilon^i$ where $\epsilon$ is a $n$-th root of unity, and proceeds from this point on. I do not understand the basis of this assumption, and if someone can explain just that I would be most obliged.
Best Answer
Your $C_n$ is a circulant matrix. All $n\times n$ complex circulant matrices share a common set of eigenvectors, and there is an explicit formula for their eigenvalues (see Wikipedia). In your specific case, the eigenvalues are $\omega_j+\omega_j^{n-1}$ for $j=0,1,2,\ldots,n-1$, where $\omega_j=e^{2\pi ij/n}$.