The inertia of Hermitian block matrices with zero main diagonal (Q1567550)

The inertia of a Hermitian block matrix \(H\) is characterized when \(H\) has a \(3\times 3\) block decomposition with null blocks on the main diagonal. Given nonnegative integers \(n_i\), \(i=1,2,3\) and the numbers \(r_{ij}\), \(R_{ij}\), \(i<j\leq 3\) such that \(0\leq r_{ij}\leq R_{ij} \leq\min \{n_i,n_j\}\), equivalent conditions with the following statement (i) are emphasized: (i) There exist \(n_i\times n_j\) matrices \(X_{ij}\), \(i<j\leq 3\) such that \(r_{ij}\leq\text{rank} X_{ij}\leq R_{ij}\) and \[ H=\left [\begin{matrix} 0 & X_{12} & X_{13} \\ X^*_{12} & 0 & X_{23}\\ X^*_{13} & X^*_{23} & 0\end{matrix} \right] \] has inertia \((\pi,\nu,*)\).