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

    Fingerprint

    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

    Lins, Brian ; Spitkovsky, Ilya ; Zhong, Siyu. / The normalized numerical range and the Davis–Wielandt shell. In: Linear Algebra and Its Applications. 2018 ; Vol. 546. pp. 187-209.
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