A quantitative formulation of Sylvester's law of inertia. III

Abstract

Using elementary matrix algebra we establish the following theorems: (1.3) Let H be any n× n Hermitian matrix and let M be any n× m complex matrix. Suppose that (i) M ∗ M has r eigenvalues in the interval [ a 1, b 1]; (ii) H has s eigenvalues in [ a 2, b 2, a 2⩾0. Then M ∗ HM has at least r+ s− n eigenvalues in [ a 1 a 2, b 1 b 2]. (3.1) Let H be any n× n Hermitian matrix with In H=(π, ν, δ). Let M be any real n× m matrix, and let δ M =DimKer M. Let (π 1, ν 1, δ 1) denote the inertia of M ∗ HM. Then π+(m −n)−δ M⩽π 1⩽π and ν+(m−n)−δ M⩽ν 1⩽ ν . When M is a square matrix, these inequalities are simply π−δ M⩽π 1⩽π and ν−δ M⩽ν 1⩽ν .