We present a new proof of the inertia result associated with Lyapunov equat
ions. Furthermore, we present a connection between the Lyapunov equation an
d the Lanczos process which is closely related to the Schwarz form of a mat
rix. We provide a method for reducing a general matrix to Schwarz form in a
finite number of steps (O(n(3))). Hence, we provide a finite method for co
mputing inertia without computing eigenvalues. This scheme is unstable nume
rically and hence is primarily of theoretical interest. (C) 2001 Elsevier S
cience Inc. All rights reserved.