6. Condition Numbers
Society for Industrial and Applied Mathematics eBooks · 2005
Let Bn be an n × n matrix. For 1 ≤ p ≤ ∞, we denote by κp(Bn) the condition number of Bn as an operator on ℓnp: κp ( Bn ) ≔ ‖ Bn ‖p ‖ B n −1 ‖p . Throughout what follows we put ∥Bn−1∥p = ∞ in case Bn is not invertible. In this chapter, we study the behavior of the condition numbers of Toeplitz band matrices Tn(b) and of Toeplitz-like matrices Bn = Tn(b) + PnK Pn + WnLWn for large n. 6.1 Asymptotic Inverses of Toeplitz-Like Matrices As in Section 5.4, we assume that Bn is given by Bn = Tn (b)+ Pn K Pn + Wn L Wn , 6.1 where b(t) = bjtj (t ∈ T) and where K and L have only a finite number of nonzero entries, that is, P n0 K P n0 =K, P n0 L P n0 =L 6.2 for some n0 ∈ N. Theorem 3.15 gives an asymptotic inverse for the pure Toeplitz matrices Tn (b). The purpose of this section is to extend this result to the Toeplitz-like matrices Bn. We put B=T (b) +K, B ∼ =T ( b ∼ ) +L, 6.3 and in case B and B∼ are invertible (which, by Theorem 1.9, implies that b is invertible and that wind b = 0), we set X= B−1 −T ( b−1 ), Y= B ∼ −1 −T ( b ∼ −1 ) . 6.4