An inertia formula for Hermitian matrices with sparse inverses

Abstract Let A ∈ Mn be a nonsingular Hermitian matrix, let G be a chordal graph on vertices {1,…,n}, and suppose that G(A−1) ⊆ G. Then the inertia of A may be expressed in a simple way in terms of the inertias of those principal submatrices of A corresponding to certain readily identifiable sets of vertices of G. Specifically, we prove that the inertia i(A) satisfies the identity i(A)= ∑ α∈C i(A[α])− ∑ β∈S m(β)i(A[β]) , in which C denotes the collection of maximal cliques of G, S denotes the collection of minimal vertex separators of G, and m(β) is the multiplicity of a separator β.