Elsevier

Linear Algebra and its Applications

Volume 390, 1 October 2004, Pages 75-109
Linear Algebra and its Applications

On flat portions on the boundary of the numerical range

https://doi.org/10.1016/j.laa.2004.04.009Get rights and content
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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

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Research of both authors supported in part by NSF Grant DMS-99-88579.