The Generalized Numerical Range of a Set of Matrices

Pan Shun Lau; Chi Kwong Li; Yiu Tung Poon; Nung Sing Sze

doi:10.1016/j.laa.2018.09.023

The Generalized Numerical Range of a Set of Matrices

Pan Shun Lau, Chi Kwong Li, Yiu Tung Poon, Nung Sing Sze

Research output: Journal article publication › Journal article › Academic research › peer-review

1 Citation (Scopus)

Abstract

For a given set of n×n matrices F, we study the union of the C-numerical ranges of the matrices in the set F, denoted by W_C(F). We obtain basic algebraic and topological properties of W_C(F), and show that there are connections between the geometric properties of W_C(F) and the algebraic properties of C and the matrices in F. Furthermore, we consider the starshapedness and convexity of the set W_C(F). In particular, we show that if F is the convex hull of two matrices such that W_C(A) and W_C(B) are convex, then the set W_C(F) is star-shaped. We also investigate the extensions of the results to the joint C-numerical range of an m-tuple of matrices.

Original language	English
Pages (from-to)	24-46
Number of pages	23
Journal	Linear Algebra and Its Applications
Volume	563
DOIs	https://doi.org/10.1016/j.laa.2018.09.023
Publication status	Published - 15 Feb 2019

Keywords

Convex set
Numerical range
Star-shaped set

ASJC Scopus subject areas

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

More information

10.1016/j.laa.2018.09.023

http://hdl.handle.net/10397/98595

Cite this

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title = "The Generalized Numerical Range of a Set of Matrices",

abstract = "For a given set of n×n matrices F, we study the union of the C-numerical ranges of the matrices in the set F, denoted by WC(F). We obtain basic algebraic and topological properties of WC(F), and show that there are connections between the geometric properties of WC(F) and the algebraic properties of C and the matrices in F. Furthermore, we consider the starshapedness and convexity of the set WC(F). In particular, we show that if F is the convex hull of two matrices such that WC(A) and WC(B) are convex, then the set WC(F) is star-shaped. We also investigate the extensions of the results to the joint C-numerical range of an m-tuple of matrices.",

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T1 - The Generalized Numerical Range of a Set of Matrices

AU - Lau, Pan Shun

AU - Li, Chi Kwong

AU - Poon, Yiu Tung

AU - Sze, Nung Sing

PY - 2019/2/15

Y1 - 2019/2/15

N2 - For a given set of n×n matrices F, we study the union of the C-numerical ranges of the matrices in the set F, denoted by WC(F). We obtain basic algebraic and topological properties of WC(F), and show that there are connections between the geometric properties of WC(F) and the algebraic properties of C and the matrices in F. Furthermore, we consider the starshapedness and convexity of the set WC(F). In particular, we show that if F is the convex hull of two matrices such that WC(A) and WC(B) are convex, then the set WC(F) is star-shaped. We also investigate the extensions of the results to the joint C-numerical range of an m-tuple of matrices.

AB - For a given set of n×n matrices F, we study the union of the C-numerical ranges of the matrices in the set F, denoted by WC(F). We obtain basic algebraic and topological properties of WC(F), and show that there are connections between the geometric properties of WC(F) and the algebraic properties of C and the matrices in F. Furthermore, we consider the starshapedness and convexity of the set WC(F). In particular, we show that if F is the convex hull of two matrices such that WC(A) and WC(B) are convex, then the set WC(F) is star-shaped. We also investigate the extensions of the results to the joint C-numerical range of an m-tuple of matrices.

KW - Convex set

KW - Numerical range

KW - Star-shaped set

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DO - 10.1016/j.laa.2018.09.023

M3 - Journal article

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SN - 0024-3795

VL - 563

SP - 24

EP - 46

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

The Generalized Numerical Range of a Set of Matrices

Abstract

Keywords

ASJC Scopus subject areas

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