Conditioning of the Stable, Discrete-Time Lyapunov Operator

MICHAEL K. TIPPETTy, STEPHEN E. COHNz, RICARDO TODLINGx, AND DAN MARCHESIN{ Abstract. The Schatten p-norm condition of the discrete-time Lyapunov operator LA defined on matrices P 2 Rn n by LAP P APAT is studied for stable matrices A 2 Rn n . Bounds are obtained for the norm of LA and its inverse that depend on the spectrum, singular values and radius of stability of A. Since the solution P of the the discrete-time algebraic Lyapunov equation (DALE) LAP = Q can be ill-conditioned only when either LA or Q is ill-conditioned, these bounds are useful in determining whether P admits a low-rank approximation, which is important in the numerical solution of the DALE for large n.