There is a Toeplitz matrix of the following form:
\begin{equation} M = \begin{pmatrix} 1 & e^{i\phi} & e^{2i\phi} & \ldots & e^{(N-1)i\phi} \\ e^{i\phi} & 1 & e^{i\phi} & \ldots & e^{(N-2)i\phi} \\ e^{2i\phi} & e^{i\phi} & 1 & \ldots & e^{(N-3)i\phi} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ e^{(N-1)i\phi} & e^{(N-2)i\phi} & e^{(N-3)i\phi} & \ldots & 1 \end{pmatrix}. \end{equation}
where $\phi$ is just some real number. This is a complex symmetric Toeplitz matrix. There is no general algorithm to diagonalize it analytically, and I am not absolutely sure that this is possible. Right now I am in the process of obtaining the eigenvalues (or at least a simple form of the eigenvalue equation).
However, what I do know is the answer in the case of $\phi=0$, then it is just a matrix of all $1$'s, which is a circulant type, and can be diagonalized by the DFT matrix. So, I wonder, may be there is some sort of simple transformation between the $\phi=0$, and $\phi \ne 0$ cases so that I can find the eigenvectors without explicitly finding the Kernel for each $M-\lambda_jI$?
Background
Matrix $M$ describes a certain periodic quantum-mechanical system. Physically, it is somewhat similar to what is called a photonic crystal, where each subsystem interacts with others by means of plane waves (hence the constant amplitude of all entries, and an acquired multiple of $\phi$ in phase).
Update 23.03.2021 So, basically, I was able to obtain an explicit form of the eigenvalue equation: \begin{equation} \det \left(M - \lambda I \right) = \dfrac{\left( \beta_+^N - \beta_-^N \right) (1-\lambda) + (\beta_-^N \beta_+ - \beta_+^N \beta_-)}{\beta_+-\beta_-} = 0, \\ \beta_{\pm} = \frac{1}{2} \left( 1-\lambda - e^{i2\phi}(1+\lambda) \pm \sqrt{ (1-\lambda - e^{i2\phi} (1+\lambda))^2 - 4 e^{i2\phi} \lambda^2 } \right), \end{equation} which is... fairly complicated. I believe this can not be solved explicitly, and, probably, the eigenvectors can not be expressed exactly. However, there is also a way to solve it by using a physical rationale, mb I will share some materials on that later.
Computing the eigensystem with Mathematica indicates that there is nothing particularly enlightening going on. Perhaps in the case where $\exp(i \theta)$ is a root of unity, something may be doable, but in general...