Do the matrices of the form $\left(\begin{matrix} \frac{\xi + \xi^{-1}}{2} & \frac{\xi - \xi^{-1}}{2} \\ \frac{\xi - \xi^{-1}}{2} & \frac{\xi + \xi^{-1}}{2} \end{matrix}\right)$ where $\xi$ is a root of unity has a particular name?
Those matrices are popping up in something I am working with and I am wondering if they have a particular name. (I am interested in things like their order and stuff.)
If $\xi=e^{i\theta}$, then your matrix is $$ \left(\begin{array}{cc}\cos\theta&i\sin\theta\\i\sin\theta&\cos\theta\end{array}\right). $$ This is similar to the matrix representing a plane rotation - conjugated by $diag(1,i)$. This time the eigenvectors are $(1,\pm1)^T$ instead of the usual $(1,\pm i)^T$. The eigenvalues are $e^{\pm i\theta}$ anyway.
If $\theta=2\pi k/n$ with $\gcd(k,n)=1$ this matrix is of order $n$.