Restricted quadratic forms, inertia theorems, and the Schur complement

Abstract

The starting point of this investigation is the properties of restricted quadratic forms x ⊤ Ax, x ϵ S ⊂ R m , where A is an m × m real symmetric matrix, and S is a subspace. The index theory of Hestenes (1951) and Maddocks (1985) that treats the more general Hilbert-space version of this problem is first specialized to the finite-dimensional context, and appropriate extensions, valid only in finite dimensions, are made. The theory is then applied to obtain various inertia theorems for matrices and positivity tests for quadratic forms. Expressions for the inertias of divers symmetrically partitioned matrices are described. In particular, an inertia theorem for the generalized Schur complement is given. The investigation recovers, links, and extends several, formerly disparate, results in the general area of inertia theorems.