 
Summary: Lyapunov, Lanczos, and Inertia y
A.C. Antoulas z and D.C. Sorensen x
October 19, 2000
Abstract. 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
nite number of steps (O(n 3 )). Hence, we provide a nite method for computing inertia without computing
eigenvalues. This scheme is unstable numerically and hence is primarily of theoretical interest.
Keywords: Lanczos method, Lyapunov Equation, Inertia, Stability, Control
AMS subject classications. Primary 65F15, Secondary 65G05
1 Introduction
The Lyapunov equation
AP+PA = M (1)
is important in linear system and control theory. Its solution has generated a lot of activity, for example
[5, 4, 6, 9, 16, 18], to name but a few. There are well known connections between system theory, particularly
linear time invariant systems, and the Lanczos method for reducing a general matrix to tridiagonal form
[12, 19, 7, 10, 11, 13, 15]; see also [14], [3]. In this paper, we present a way to use the Lanczos method to
solve the Lyapunov equation directly using O(n 3 )
oating point operations without computing eigenvalues.
This scheme is mainly of theoretical interest since it is numerically unstable.
In addition to solving the Lyapunov equation, a slight modication of this scheme has the ability to
