Finding the root of the matrix representation of a cyclic permutation of lenght $n$

Question

I am wondering if there is a general formula for the matrix square root of the representation $C_n$ of a cyclic permutation of period $n$. The matrix $A$ such that $A^2=C_n$. For example $n=3$: $$ A^2 = \begin{pmatrix} 0 &0&1\\ 1&0&0\\ 0&1&0 \end{pmatrix}=C_3 $$

Answer 1

The equation $A^2=C_3$ has two obvious solutions with left-upper corner equal to zero, which are even integral, namely $$ A=\begin{pmatrix} 0 & -1 & 0 \cr 0 & 0 & -1 \cr -1 & 0 & 0 \end{pmatrix},\; A=\begin{pmatrix} 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \end{pmatrix}. $$ In general, there are several articles on squares roots of circulant matrices.