Exponential Condition Number of Solutions of the Discrete Lyapunov Equation

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 examine