The matrix M described below has double root eigenvalues. Could you tell me if there are any prior research theories on this matter? Here, $i=\sqrt{-1}$.
\begin{equation} M= \frac{1}{2} \quad \left( \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & i & -1 & -1 \\ 1 & -1 & 1 & 1 \\ 1 & -1 & -1 & i \end{array} \right) \end{equation} The eigenvalue equation for this is as follows. \begin{equation} |\lambda I_4 - M | = \lambda(\lambda -1)( \lambda -i )^2 \end{equation} I understand that everyone may be busy, but I would greatly appreciate it if you could provide me with the information.
Supplementary Explanations
At first glance, it seems that this matrix possesses distinct eigenvalues. Nevertheless, due to the calculated outcome resulting in multiple solutions, it is anticipated that these eigenvalues are distributed along the unit circle. In essence, we are inquiring into the possibility, positing a hypothesis, that they could potentially be expressed through trigonometric functions. It's important to note that the question is devoid of explicit context for the sake of clarity.
Amendment
\sout{I believe that regarding M as Complex Hadamard Matrix is more appropriate than considering it as Vandermonde matrix.}