The Gau–Wu number for 4 × 4 and select arrowhead matrices

Kristin A. Camenga; Patrick X. Rault; Ilya M. Spitkovsky; Rebekah B.Johnson Yates

doi:10.1016/j.laa.2022.02.026

The Gau–Wu number for 4 × 4 and select arrowhead matrices

Kristin A. Camenga, Patrick X. Rault, Ilya M. Spitkovsky, Rebekah B.Johnson Yates

Mathematics

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

Abstract

The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number k(A) of orthonormal vectors x_j such that the scalar products 〈Ax_j,x_j〉 lie on the boundary of the numerical range of A) is computed for a class of arrowhead matrices A of arbitrary size, including dichotomous ones. These results are then used to completely classify all 4×4 matrices according to the values of their Gau–Wu numbers.

Original language	English (US)
Pages (from-to)	192-218
Number of pages	27
Journal	Linear Algebra and Its Applications
Volume	644
DOIs	https://doi.org/10.1016/j.laa.2022.02.026
State	Published - Jul 1 2022

Keywords

4×4 matrices
Arrowhead matrix
Boundary generating curve
Field of values
Gau–Wu number
Irreducible
Numerical range
Singularity

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.2022.02.026

Cite this

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title = "The Gau–Wu number for 4 × 4 and select arrowhead matrices",

abstract = "The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number k(A) of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range of A) is computed for a class of arrowhead matrices A of arbitrary size, including dichotomous ones. These results are then used to completely classify all 4×4 matrices according to the values of their Gau–Wu numbers.",

keywords = "4×4 matrices, Arrowhead matrix, Boundary generating curve, Field of values, Gau–Wu number, Irreducible, Numerical range, Singularity",

author = "Camenga, {Kristin A.} and Rault, {Patrick X.} and Spitkovsky, {Ilya M.} and Yates, {Rebekah B.Johnson}",

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doi = "10.1016/j.laa.2022.02.026",

language = "English (US)",

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pages = "192--218",

journal = "Linear Algebra and Its Applications",

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T1 - The Gau–Wu number for 4 × 4 and select arrowhead matrices

AU - Camenga, Kristin A.

AU - Rault, Patrick X.

AU - Spitkovsky, Ilya M.

AU - Yates, Rebekah B.Johnson

PY - 2022/7/1

Y1 - 2022/7/1

N2 - The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number k(A) of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range of A) is computed for a class of arrowhead matrices A of arbitrary size, including dichotomous ones. These results are then used to completely classify all 4×4 matrices according to the values of their Gau–Wu numbers.

AB - The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number k(A) of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range of A) is computed for a class of arrowhead matrices A of arbitrary size, including dichotomous ones. These results are then used to completely classify all 4×4 matrices according to the values of their Gau–Wu numbers.

KW - 4×4 matrices

KW - Arrowhead matrix

KW - Boundary generating curve

KW - Field of values

KW - Gau–Wu number

KW - Irreducible

KW - Numerical range

KW - Singularity

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DO - 10.1016/j.laa.2022.02.026

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SN - 0024-3795

VL - 644

SP - 192

EP - 218

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

The Gau–Wu number for 4 × 4 and select arrowhead matrices

Abstract

Keywords

ASJC Scopus subject areas

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Cite this

The Gau–Wu number for 4 × 4 and select arrowhead matrices

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this

The Gau–Wu number for 4 × 4 and select arrowhead matrices