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Structured perturbations and symmetric matrices
Publikationstyp
Journal Article
Date Issued
2000-06-20
Sprache
English
Author(s)
Institut
TORE-URI
Volume
278
Issue
1-3
Start Page
121
End Page
132
Citation
Linear Algebra and Its Applications 278 (1-3): 121-132 (1998)
Publisher DOI
Scopus ID
Publisher
American Elsevier Publ.
For a given n × n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal Bauer-Skeel condition number cannot be larger than (3 + 2√2)n. In this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation, for relative perturbations of the individual matrix components. It is shown that a symmetric linear system can be arbitrarily ill-conditioned, while any symmetric and entrywise relative perturbation of the matrix of less than 100% does not produce a singular matrix. That means that the inverse of the condition number and the distance to the nearest ill-posed problem can be arbitrarily far apart. Finally we prove that restricting structured perturbations to symmetric (entrywise) perturbations cannot change the condition number by more than a factor (3 + 2\√2)n.
Subjects
Condition number
Structured perturbations
Symmetric matrices
DDC Class
510: Mathematik