Conditioning of the Stable, Discrete-time Lyapunov Operator

The Schatten p-norm condition of the discrete-time Lyapunov operator L(sub A) defined on matrices P is identical with R(sup n X n) by L(sub A) P is identical with P - APA(sup T) is studied for stable matrices A is a member of R(sup n X n). Bounds are obtained for the norm of L(sub A) 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) L(sub A)P = Q can be ill-conditioned only when either L(sub 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.