The inertia of certain Hermitian skew-triangular block matrices

AbstractLet n1, n2, n3 be nonnegative integers. We consider Hermitian matrices H of the form H=H11H12H13H21H22H23H31H32H33 where each Hij is ni × nj. We characterize the set of inertias {In(H): In(Hii) = (πi, vi, δi) and r1, i + 1 ⩽ rank H1, i + 1 ⩽ R1, i + 1 for i = 1, 2} in terms of π1, v1, δ1, π2, v2, δ2, n3, r12, r13, R12, R 12, and we discuss the implications of this characterization for the determination of the inertia of other types of Hermitian skew-triangular block matrices