The inertia of a Hermitian matrix having prescribed diagonal blocks

For i=1,…,m let H_i be an n_i×n_i Hermitian matrix with inertia In(H_i)= (π_i, ν_i, δ_i). We characterize in terms of the π_i, ν_i, δ_i the range of In(H) where H varies over all Hermitian matrices which have a block decomposition H= (X_ij)_{i, j=1,…, m} in which X_ij is n_i×n_j and X_ii=H_i.