Exponential Condition Number of Solutions of the Discrete Lyapunov Equation

The condition number of the n×n matrix P is examined, where P solves P-APA*=BB*, and B is a n×d matrix. Lower bounds on the condition number κ 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 (IN) form. A simulation shows that P is typically ill-conditioned in the case of n≫1 and d=1. When Aij has an independent Gaussian distribution (subject to restrictions), we observe that κ(P)1n/∼3.3. The effect of autocorrelated forcing on the conditioning on state space systems is examined.