Eigenvalue patterned condition numbers: Toeplitz and Hankel cases

Abstract

We continue the study started in [Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174-189] concerning the sensitivity of simple eigenvalues of a matrix A to perturbations in A that belong to a chosen subspace of matrices. In [Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174-189] the zero-structured perturbations have been considered. Here we focus on patterned perturbations, and the cases of the Toeplitz and of the Hankel matrices are investigated in detail. Useful expressions of the absolute patterned condition number of the eigenvalue lambda and of the analogue of the matrix yx(H), which leads to the traditional condition number of)., are given. MATLAB codes are defined to compare traditional, zero-structured and patterned condition numbers. A report on significant numerical tests is included. (C) 2006 Elsevier B.V. All rights reserved

Structured Pseudospectra for Small Perturbations

“…The latter have been defined and investigated in [27]. For structured condition numbers of simple eigenvalues, see, e.g., [5,6,8,9,16,22,23,29,30,37,38]. Finally, we apply our results to the case of real perturbations of real matrices.…”

Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

“…Relation (6.3) is Tisseur's formula [23, section 4] in another notation. From (6.3) one obtains the results of Noschese and Pasquini [18] on the structured condition number of Toeplitz matrices by observing that the orthogonal projection of a matrix X ∈ C n×n onto the set of Toeplitz matrices is given by replacing the entries in each diagonal of X by their arithmetic mean.…”

“…It measures the sensitivity of the eigenvalue λ if the matrix A is subjected to perturbations from the class Δ. In recent years some work has been done in order to obtain estimates or computable formulae for κ Δ (A, λ) [3,4,5,7,13,15,16,18,17,20,21,23]. However, the condition number cannot reveal how the eigenvalue moves in a specific direction under structured perturbations.…”

On componentwise condition numbers for eigenvalue problems with structured matrices

“…Condition numbers for eigenvalue computations for several classes of structured matrices have been studied in [6][7][8][9][10] (for normwise condition numbers) and in [10] (for componentwise condition numbers and the classes of circulant, symmetric, Hermitian and skew-Hermitian matrices). Higham and Higham [11] studied structured backward and condition of generalized eigenvalue problems and developed the approach based on the Kronecker product, which will be used in this paper.…”