Say $M=M^{*}$ is an n*n Hermitian matrix with rank = r. The question is how to determine the signature of the $2n*2n$ block matrix \begin{matrix} 0 & M \\ M & 0 \end{matrix}
If we start with small n, it is apparent we can find the pattern that the signature should be things like (r,r,0), however, I'm not sure exactly what is the clearest way to show that in a general case.