I have the following problem: I have a circulating matrix $$ C= \begin{bmatrix} c_0 & c_{n-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{n-1} & & c_{2} \\ \vdots & c_{1}& c_0 & \ddots & \vdots \\ c_{n-2} & & \ddots & \ddots & c_{n-1} \\ c_{n-1} & c_{n-2} & \dots & c_{1} & c_0 \\ \end{bmatrix} $$
and I am trying to find bases for the Eigenspaces of $C$.
I have already shown several things: For a matrix $A$ with $$ \quad A= \begin{pmatrix} 0 & 0 & \dots & 1 \\ 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \dots & 0\\ \end{pmatrix} \in \mathbb C^{n \times n}$$
I have shown that the eigenvalues $t_j$ of $A$ are $t,t^2,t^3, \dots , t^n$ with $t= e^{i\phi}, \ \phi = \frac{2\pi}{n}$. I have also found that the eigenvectors of $A$ must then be: $(1,t^k,t^{2k},...,t^{(n-1)k}), k\in \{1,...,n\}$ .
Now I know that I can write $C$ as a linear combination of $A$, such that $$C = c_0+c_1A+...+c_{n-1}A^{n-1}$$
Let $Q(A)$ denote this polynomial. I have also shown that I can write $Q(A)=\sum_{j=1}^nQ(t_j)p_j$ whereas $t_j$ denote the eigenvalues of $A$ and $p_j$ denote the orthogonal projection onto the $j$-th canonical basis vector.
Therefore I know that the eigenvalues of $C$ must be $Q(t_j)$, i.e $\lambda_j= c_0+c_1t_j+...c_{n-1}t_j^{n-1}$
Now I am stuck however. I am trying to find a basis for the eigenspaces of $C$, which should be easily doable using what I have already shown, but I am not quite sure what I could do.
Any hints/help would be greatly appreciated!
Hint: If $v_j$ is an eigenvector of $A$ associated with $t_j$, then it is also an eigenvector of $Q(A)$ associated with $Q(t_j)$.