The normalized numerical range and the Davis–Wielandt shell

Brian Lins, Ilya M. Spitkovsky, Siyu Zhong

Research output: Contribution to journalArticlepeer-review


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
StatePublished - Jun 1 2018


  • 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


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