On the inertias of symmetric matrices and bounded self-adjoint operators

AbstractLet S be a Hermitian matrix, and S1 a principal submatrix with r less rows and columns. Let π and π1 be the numbers of positive eigenvalues of S and S1. One of Poincaré's inequalities says that π1 + r ⩾ π ⩾ π1. We tighten these inequalities as follows. Let δ and δ1 be DimKerS and DimKerS1, and let d = DimKerS ∩ KerS1. Then π1 + r − (δ − d) ⩾ π ⩾ π1 + (δ1 − d)