Conditioning of the Stable, Discrete-Time Lyapunov Operator

Abstract

The Schatten p-norm condition of the discrete-time Lyapunov operator $\A$ defined on matrices $P \in \mathbb{R}^{n \times n}$ by $\A P \equiv P -A P A ^{T}$ is studied for stable matrices $A \in \mathbb{R}^{n \times n}$. Bounds are obtained for the norm of $\A$ 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) $\A P = Q$ can be ill-conditioned only when either $\A$ 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.