Exponential Condition Number of Solutions of the Discrete Lyapunov Equation

TL;DR

This paper investigates how the condition number of solutions to the discrete Lyapunov equation grows exponentially with system size, providing bounds and empirical insights into ill-conditioning in high-dimensional systems.

Contribution

It derives lower bounds on the condition number for solutions of the discrete Lyapunov equation under various matrix structures and explores the effects of random and auto-correlated inputs.

Findings

Condition number grows exponentially with system dimension n.

Solutions are typically ill-conditioned for large n, especially when d=1.

Empirical results show a typical condition number growth rate of approximately 3.3 per dimension.

Abstract

The condition number of the $n\ x\ n$ matrix $P$ is examined, where $P$ solves %the discete Lyapunov equation, $P - A P A^* = BB^*$, and $B$ is a $n\ x\ d$ matrix. Lower bounds on the condition number, $\kappa$, of $P$ are given when $A$ is normal, a single Jordan block or in Frobenius form. The bounds show that the ill-conditioning of $P$ grows as $\exp(n/d) >> 1$. These bounds are related to the condition number of the transformation that takes $A$ to input normal form. A simulation shows that $P$ is typically ill-conditioned in the case of $n>>1$ and $d=1$. When $A_{ij}$ has an independent Gaussian distribution (subject to restrictions), we observe that $\kappa(P)^{1/n} ~= 3.3$. The effect of auto-correlated forcing on the conditioning on state space systems is examined