Consider the $n \times n$ Toeplitz matrix
\begin{equation} T_n = \begin{bmatrix} a & b & 0 & 0 & \cdots & 0 \\ \bar{b} & a & b & 0 & \cdots & 0\\ 0 & \bar{b} & a & b & \cdots & 0\\ \vdots &\vdots & \ddots &\ddots &\ddots &\vdots \\ 0 & 0 & 0 & 0 & \bar{b} &a \end{bmatrix}. \end{equation} I can calculate the eigenvalues which would be
\begin{equation} a + 2|b| \cos \frac{k \pi}{n+1} \quad 1\le k \le n. \end{equation} This follows from simple Fourier series calculation. However, how to calculate the eigenvectors? Is there a general formula as well?
In general what happens for eigenvectors of Hermitian Toeplitz operators? Advanced help for any help/ suggestion/ references.