On componentwise condition numbers for eigenvalue problems with structured matrices

Abstract

We give explicit expressions for the componentwise condition number for eigenvalue problems with structured matrices. We will consider only linear structures and show a general result from which expressions for the condition numbers follow. We obtain explicit expressions for the following structures: Toeplitz and Hankel. Details for other linear structures should follow in a straightforward manner from our general result.We mentioned above that the expressions for componentwise condition numbers depend on the derivative D of . The goal of this section is to make explicit this dependance. We do so in Lemma 4, which follows an idea from [12]. We begin by recalling that, for a matrix A ∈ R ×k , the

The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix

“…The effect of structurepreserving perturbations of the matrix entries has been analyzed in the context of linear systems (see, e.g., [14,29,30,35]) as well as for structured eigenproblems. Measures of structured eigenvalue sensitivity include both structured pseudospectra (see, e.g., [2,6,12,16,24,31]) and structured condition numbers, introduced by Tisseur [34]. The latter measures have been investigated, e.g., in [17,21,23] and, recently, applied to network analysis [9,26].…”

The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix

“…The effect of structure-preserving perturbations of the matrix entries has been analyzed in the context of linear systems (see, e.g., [37,15,31,32]) as well as for structured eigenproblems. Measures of structured eigenvalue sensitivity include both structured pseudospectra (see, e.g., [11,30,7,17,3,25]) and structured condition numbers, introduced by Tisseur [35]. The latter measures have been investigated, e.g., in [18,22,24,2] and, recently, applied to network analysis [28,10].…”