On flat portions on the boundary of the numerical range

Ethan S. Brown; Ilya M. Spitkovsky

doi:10.1016/j.laa.2004.04.009

On flat portions on the boundary of the numerical range

Ethan S. Brown, Ilya M. Spitkovsky

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The paper is devoted to matrices with flat portions on the boundary of their numerical range. A constructive criterion for such portions to exist is obtained in case of tridiagonal matrices, and a particular case of continuant matrices is considered. As an application, the cases of (arbitrary) 3×3 and 4×4 matrices are treated. It is shown, in particular, that the sharp bound for the number of flat portions on the boundary of the numerical range for 4×4 matrices is four (three, if the matrices are assumed unitarily irreducible).

Original language	English (US)
Pages (from-to)	75-109
Number of pages	35
Journal	Linear Algebra and Its Applications
Volume	390
Issue number	1-3
DOIs	https://doi.org/10.1016/j.laa.2004.04.009
State	Published - Oct 1 2004

Keywords

Numerical range
Tridiagonal matrices
Unitary (ir)reducibility

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2004.04.009

Cite this

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title = "On flat portions on the boundary of the numerical range",

abstract = "The paper is devoted to matrices with flat portions on the boundary of their numerical range. A constructive criterion for such portions to exist is obtained in case of tridiagonal matrices, and a particular case of continuant matrices is considered. As an application, the cases of (arbitrary) 3×3 and 4×4 matrices are treated. It is shown, in particular, that the sharp bound for the number of flat portions on the boundary of the numerical range for 4×4 matrices is four (three, if the matrices are assumed unitarily irreducible).",

keywords = "Numerical range, Tridiagonal matrices, Unitary (ir)reducibility",

author = "Brown, {Ethan S.} and Spitkovsky, {Ilya M.}",

note = "Funding Information: Research of both authors supported in part by NSF Grant DMS-99-88579. ∗ Corresponding author. E-mail addresses: esbrown@mit.edu (E.S. Brown), ilya@math.wm.edu (I.M. Spitkovsky). URL: http://www.math.wm.edu/∼ilya (I.M. Spitkovsky).",

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N2 - The paper is devoted to matrices with flat portions on the boundary of their numerical range. A constructive criterion for such portions to exist is obtained in case of tridiagonal matrices, and a particular case of continuant matrices is considered. As an application, the cases of (arbitrary) 3×3 and 4×4 matrices are treated. It is shown, in particular, that the sharp bound for the number of flat portions on the boundary of the numerical range for 4×4 matrices is four (three, if the matrices are assumed unitarily irreducible).

AB - The paper is devoted to matrices with flat portions on the boundary of their numerical range. A constructive criterion for such portions to exist is obtained in case of tridiagonal matrices, and a particular case of continuant matrices is considered. As an application, the cases of (arbitrary) 3×3 and 4×4 matrices are treated. It is shown, in particular, that the sharp bound for the number of flat portions on the boundary of the numerical range for 4×4 matrices is four (three, if the matrices are assumed unitarily irreducible).

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KW - Tridiagonal matrices

KW - Unitary (ir)reducibility

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On flat portions on the boundary of the numerical range

Abstract

Keywords

ASJC Scopus subject areas

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