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# Almost sharp bounds for the componentwise distance to the nearest singular matrix

Publikationstyp

Journal Article

Publikationsdatum

1997

Sprache

English

Author

Institut

TORE-URI

Enthalten in

Volume

42

Issue

2

Start Page

93

End Page

107

Citation

Linear and Multilinear Algebra 42 (2): 93-107 (1997)

Publisher DOI

Scopus ID

Publisher

Taylor & Francis

The normwise distance of a regular matrix A ∈ M (ℝ) to the nearest singular matrix is well known to be ∥A ∥ . Such a normwise distance neglects small entries in the matrix, and it does not allow for weights in a perturbation. The reciprocal ∥ |A |·E∥ of the Bauer-Skeel condition number is known to be a lower bound for the componentwise distance of A to the nearest singular matrix weighted by the nonnegative matrix E. In this paper we derive an upper bound for this componentwise distance involving the Bauer-Skeel condition number. We show that this upper bound is sharp up to a constant factor less than 3 + 2√2, independent of A and E. For finite values of n, improved constants are given as well. © 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands under license by Gordon and Breach Science Publishers. n -1 -1 -1 -1