13. Structured Perturbations
Society for Industrial and Applied Mathematics eBooks · 2005
The pseudospectrum spεTn(b) measures the extent to which the spectrum of Tn(b) may change by an arbitrary perturbation of norm at most ε. We now consider perturbations that have the same Toeplitz band structure as Tn (b). In this way we can find the distance of Tn(b) to the nearest singular matrix within the set of all matrices of the same banded structure as Tn(b). Various condition numbers measure the sensitivity of properties of a matrix subject to perturbations. If the matrix has a certain structure, it is natural to require that the perturbations be of the same structure. This leads to the notion of structured condition numbers. In this chapter we study structured condition numbers for the Toeplitz structure. We give in particular a probabilistic argument which shows that in general we do not win anything by passing from unstructured condition numbers of banded Toeplitz matrices to Toeplitz-structured condition numbers. 13.1 Toeplitz Pseudospectra Let denote the set of all Laurent polynomials c of the form c(t) = ∑j=−rs cjtj (t ∈ T). For and ε > 0, we define the Toeplitz-structured pseudospectrum spεToep[r,s]Tn(b) by sp ε Toep [r,s] Tn (b) = ∪ φ∈ Pr,s , ‖φ‖∞ ≤ε sp Tn (b+φ) . Clearly, spεToep[r,s]Tn (b) is a subset of spε Tn(b). We denote by Uσ (λ0) the open disk of radius σ centered at λ0. Lemma 13.1. Let c If λ0 ∉ Λ(c), then there exist n0 ∈ N, σ > 0, δ > 0 such that Uσ(λ0) ∩ sp Tn(c + φ) = ∅ whenever ∣φ∣∞ ≤ δ, and n ≥ n0. Proof. From Theorem 11.3 we infer that there is a number ϱ ∈ (0, ∞) such that λ0 does not belong to sp T(cϱ).