Consider a self-adjoint matrix $M$ that has block form


$$M = \begin{pmatrix} M\_{11} & M\_{12} \\ M\_{12}^\* & M\_{11} \end{pmatrix}.$$


I am wondering if there exists any criterion to decide if this matrix can be transformed by some invertible matrix $T$


such that $$TMT^{-1} = \begin{pmatrix}0 & C \\ C^\* & 0 \end{pmatrix}$$
for some suitable matrix $C?$


Notice that one restriction that $\begin{pmatrix}0 & C \\ C^\* & 0 \end{pmatrix}$ already puts is that the spectrum of $M$ has to be symmetric with respect to zero as conjugation by $$\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}$$ shows.


As a first step, one might ask when we can achieve a form


$$TMT^{-1} = \begin{pmatrix}0 & C \\ D & 0 \end{pmatrix}$$


where $C$ and $D$ are arbitrary matrices?



Off-diagonalize a matrix
This is a so-called chiral symmetry. The restriction on the symmetry of the spectrum of $M$ is the only restriction you need, you can then bring $M$ to the desired off-diagonal form by a unitary transformation:
$$M=U\begin{pmatrix}\Lambda&0\\ 0&-\Lambda\end{pmatrix}U^\ast\Rightarrow \Omega^\ast U^\ast MU\Omega =\begin{pmatrix}0&\Lambda\\ \Lambda&0\end{pmatrix},$$
for $\Omega=2^{-1/2}\begin{pmatrix}1&1\\ -1 &1\end{pmatrix}$.


Here $U$ is the unitary matrix of eigenvectors of $M$; the eigenvalues are contained in the diagonal matrix $\Lambda$.


