I have a matrix of the form $D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right), $ where $C$ is not necessarily hermitian. Can we in general say anything about the eigenvalues of $D$ in terms of $C$? In particular, I'm interested in the case where every eigenvalue of $C$ has algebraic multiplicity 2, and what that implies about the multiplicities of eigenvalues of $D$.

I have a matrix of the form

$$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$

where $C$ is not necessarily hermitian. In general, can we say anything about the eigenvalues of $D$ in terms of $C$? In particular, I'm interested in the case where every eigenvalue of $C$ has algebraic multiplicity 2, and what that implies about the multiplicities of the eigenvalues of $D$.