Exponential condition number of solutions of the discrete Lyapunov equation

The condition number of the n/spl times/n matrix P is examined, where P solves P-APA/sup */=BB/sup */, and B is a n/spl times/d matrix. Lower bounds on the condition number /spl 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)/spl Gt/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/spl Gt/1 and d=1. When A/sub ij/ has an independent Gaussian distribution (subject to restrictions), we observe that /spl kappa/(P)/sup 1/n//spl sim/3.3. The effect of autocorrelated forcing on the conditioning on state space systems is examined.