Topological Aspects of Sylvester's Theorem on the Inertia of Hermitian Matrices

1. A set of nXn matrices with complex elements has a natural topology associated with it. One may therefore look for a topological interpretation of some results in the theory of matrices. We shall show that Sylvester's classical theorem on the inertia (signature) of Hermitian matrices concerns the connected components of the space of all Hermitian matrices of fixed rank r. Most of the arguments used in the proof of our theorem are elementary and familiar. Yet our result does not appear in the literature. The reason may well be that matrix theorists tend to use "continuity properties" as they arise, without formalizing them, while topologists do not usually study equivalence relations on matrices. This note is offered as an illustration that even on a fairly elementary level, something is gained by looking for inter-connections between different mathematical fields.