Inertia theory

AbstractThis cross section of inertia theory exposes, with some digressions, two main themes. The more historical one starts with the first theorems about inertia (those of Sylvester and Lyapunov), and reveals how they are now viewed and applied. Here the emphasis is on unification and generalization within the original finite dimensional setting. The Main Inertia Theorem is one of the principal achievements. The other theme treats the difficulties of developing analogous results for infinite dimensions. A variety of infinite dimensional “inertias” are defined and studied, and a counterpart to the Main Inertia Theorem is given, which is valid in general Hilbert spaces. Some generalizations of Sylvester's Theorem appear here for the first time, and they fit nicely into both themes. Even when they are restricted to finite dimensions they are extensions of what was previously known. João Filipe Queiró has written an extensive account of a rather different part of inertia theory [28]