We present a new proof of the inertia result associated with Lyapunov equations. Furthermore we present a connection between the Lyapunov equation and the Lanczos process which is closely related to the Schwarz form of a matrix. We provide a method for reducing a general matrix to Schwarz form in a finite number of steps (O(n3)). Hence, we provide a finite method for computing inertia without computing eigenvalues. This scheme is unstable numerically and hence is primarily of theoretical interest.