The normalized numerical range and the Davis–Wielandt shell

Brian Lins, Ilya Spitkovsky, Siyu Zhong

Research output: Contribution to journalArticle

Abstract

For a given n-by-n matrix A, its normalized numerical range FN(A) is defined as the range of the function fN,A:x↦(xAx)/(‖Ax‖⋅‖x‖) on the complement of ker⁡A. We provide an explicit description of this set for the case when A is normal or n=2. This extension of earlier results for particular cases of 2-by-2 matrices (by Gevorgyan) and essentially Hermitian matrices of arbitrary size (by A. Stoica and one of the authors) was achieved due to the fresh point of view at FN(A) as the image of the Davis–Wielandt shell DW(A) under a certain non-linear mapping h:R3↦C.

Original languageEnglish (US)
Pages (from-to)187-209
Number of pages23
JournalLinear Algebra and Its Applications
Volume546
DOIs
StatePublished - Jun 1 2018

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Numerical Range
Shell
Nonlinear Mapping
Hermitian matrix
Complement
Arbitrary
Range of data

Keywords

  • Davis–Wielandt shell
  • Normal matrix
  • Normalized numerical range

ASJC Scopus subject areas

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

Cite this

The normalized numerical range and the Davis–Wielandt shell. / Lins, Brian; Spitkovsky, Ilya; Zhong, Siyu.

In: Linear Algebra and Its Applications, Vol. 546, 01.06.2018, p. 187-209.

Research output: Contribution to journalArticle

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