The normalized numerical range and the Davis–Wielandt shell

Brian Lins; Ilya M. Spitkovsky; Siyu Zhong

doi:10.1016/j.laa.2018.01.027

The normalized numerical range and the Davis–Wielandt shell

Brian Lins, Ilya M. Spitkovsky, Siyu Zhong

Mathematics

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

Abstract

For a given n-by-n matrix A, its normalized numerical range F_N(A) is defined as the range of the function f_N,A:x↦(x^⁎Ax)/(‖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 F_N(A) as the image of the Davis–Wielandt shell DW(A) under a certain non-linear mapping h:R³↦C.

Original language	English (US)
Pages (from-to)	187-209
Number of pages	23
Journal	Linear Algebra and Its Applications
Volume	546
DOIs	https://doi.org/10.1016/j.laa.2018.01.027
State	Published - Jun 1 2018

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

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10.1016/j.laa.2018.01.027

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title = "The normalized numerical range and the Davis–Wielandt shell",

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↦(x⁎Ax)/(‖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.",

keywords = "Davis–Wielandt shell, Normal matrix, Normalized numerical range",

author = "Brian Lins and Spitkovsky, {Ilya M.} and Siyu Zhong",

note = "Funding Information: The results are partially based on the Capstone project of the third named under the supervision of the second named author. The latter was also supported in part by the Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

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doi = "10.1016/j.laa.2018.01.027",

language = "English (US)",

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AU - Lins, Brian

AU - Spitkovsky, Ilya M.

AU - Zhong, Siyu

N1 - Funding Information: The results are partially based on the Capstone project of the third named under the supervision of the second named author. The latter was also supported in part by the Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi. Publisher Copyright: © 2018 Elsevier Inc.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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↦(x⁎Ax)/(‖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.

AB - 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↦(x⁎Ax)/(‖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.

KW - Davis–Wielandt shell

KW - Normal matrix

KW - Normalized numerical range

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -

The normalized numerical range and the Davis–Wielandt shell

Abstract

Keywords

ASJC Scopus subject areas

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