III-conditioned matrices are componentwise near to singularity

Publikationstyp
Journal Article
Date Issued
1999-01-01
Sprache
English
Author(s)
Rump, Siegfried M.  orcid-logo
Institut
Zuverlässiges Rechnen E-19  
TORE-URI
http://hdl.handle.net/11420/9362
Journal
SIAM review  
Volume
41
Issue
1
Start Page
102
End Page
112
Citation
SIAM Review 41 (1): 102-112 (1999-01-01)
Publisher DOI
10.1137/S0036144598323216
Scopus ID
2-s2.0-0003031076
Publisher
SIAM
For a square matrix normed to 1, the normwise distance to singularity is well known to be equal to the reciprocal of the condition number. In this paper we give an elementary and self-contained proof for the fact that an ill-conditioned matrix is also not far from a singular matrix in a componentwise sense. This is shown to be true for any weighting of the componentwise distance. In other words, for matrix inversion, "ill conditioned" means "nearly ill posed" in the normwise and also in the componentwise sense. © 1999 Society for Industrial and Applied Mathematics.
Subjects
Componentwise distances
Condition number
Ill conditioned
Ill posed
Sign-real spectral radius
DDC Class
004: Informatik
510: Mathematik