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Counterexample to a conjecture of Elsner on the spectral variation of matrices

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
349
Identifiers
DOI: 10.1016/s0024-3795(01)00605-x
Keywords
  • Spectral Variation
  • Operator Norm

Abstract

Abstract In 1985, Elsner proved that the Hausdorff distance Δ between the spectra of two n× n matrices A and B satisfies Δ σ(A),σ(B) n⩽ ∥A∥+∥B∥ n−1∥A−B∥, where ∥·∥ denotes the operator norm with respect to the Euclidean norm on C n . He further conjectured that the same inequality holds for all operator norms. We disprove this conjecture, and also the weaker conjecture where (∥ A∥+∥ B∥) is replaced by 2max(∥ A∥,∥ B∥).

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