### 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 kerA. 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:R^{3}↦C.

Original language | English (US) |
---|---|

Pages (from-to) | 187-209 |

Number of pages | 23 |

Journal | Linear Algebra and Its Applications |

Volume | 546 |

DOIs | |

State | Published - Jun 1 2018 |

### Fingerprint

### 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

*Linear Algebra and Its Applications*,

*546*, 187-209. https://doi.org/10.1016/j.laa.2018.01.027

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

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 546, pp. 187-209. https://doi.org/10.1016/j.laa.2018.01.027

}

TY - JOUR

T1 - The normalized numerical range and the Davis–Wielandt shell

AU - Lins, Brian

AU - Spitkovsky, Ilya

AU - Zhong, Siyu

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 kerA. 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 kerA. 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

UR - http://www.scopus.com/inward/record.url?scp=85042199032&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042199032&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2018.01.027

DO - 10.1016/j.laa.2018.01.027

M3 - Article

VL - 546

SP - 187

EP - 209

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -