Conditioning of the stable, discrete-time Lyapunov operator

The Schatten p-norm condition of the discrete-time Lyapunov operator L-A defined on matrices P is an element of R-nxn by LAP = P - APA(T) is studied for stable matrices A is an element of R-nxn. Bounds are obtained for the norm of L-A and its inverse that depend on the spectrum, singular values, and radius of stability of A. Since the solution P of the discrete-time algebraic Lyapunov equation ( DALE) LAP = Q can be ill-conditioned only when either L-A 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.