The possible inertias for a Hermitian matrix and its principal submatrices

Abstract Definition: A Hermitian matrix H is a Hermitian extension of a given set of Hermitian matrices { H ii , i = 1,…, m } if these { H ii } are the block diagonals of H . Let (π i , v i ,δ i ) = In H ii , the inertia of each H ii . Special case: Given Hermitian matrices { H ii , i =1,…, m } and given nonnegative integers π, v , and δ such that π+ v +δ=Σ(π i + v i +δ i ); then a Hermitian extension H exists such that Ker H ⊃⊕Ker H ii and In H =(π, v , δ) if and only if δ⩾Σδ i and π⩾max π i and v ⩾max v i . We also present a simple extension theorem for the general case (Ker H ⊅ ⊕ Ker H ii ).