Exponential Condition Number of Solutions of the Discrete Lyapunov Equation
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The condition number of the n x n matrix P is examined, where P solves x 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 (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 κ(P)^1/n = 3.3. The effect of auto-correlated forcing on the conditioning on state space systems is examined
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