I have a matrix of this form
\begin{bmatrix}
\ 0 & v_1 & we^{-ik} \\[0.3em]
\ v_1 & 0 & v_2 \\[0.3em]
we^{ik} & v_2 & 0
\end{bmatrix}
where $v_1$, $v_2$, $w$ and $k$ are some parameters. Is it possible to block diagonalise this matrix without specifying the values of $v_1$, $v_2$, $w$ and $k$? I have seen some examples about diagonalising a complex matrix. It seems that I need to calculate the eigenvalues, but this matrix does not have simple analytic eigenvalues. These are the eigenvalues calculated using Mathematica:
It is also weird that there are some imaginary numbers in the eigenvalues as this matrix is clearly Hermitian.
Any help is appreciated.