THE COPOSITIVE RANGE

Seong Jun Park; Nurhan Çolakoğlu; Michael J. Tsatsomeros

doi:10.13001/ela.2025.9741

THE COPOSITIVE RANGE

Seong Jun Park, Nurhan Çolakoğlu, Michael J. Tsatsomeros^*

^*Corresponding author for this work

Department of Mathematics Engineering

Department of Mathematics and Statistics

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

Abstract

We consider symmetric copositive matrices A ∈ M_n (R), which by definition satisfy x^T Ax ≥ 0 for all nonzero x ≥ 0. We introduce the notion the copositive range of a copositive matrix A, CR(A) = {x^T Ax: x ≥ 0, ‖x‖₂ = 1}, and prove that CR(A) is an interval contained in the numerical range of A. We focus on the properties and the endpoints of CR(A), which are associated with the Pareto eigenvalues of A.

Original language	English
Pages (from-to)	575-587
Number of pages	13
Journal	Electronic Journal of Linear Algebra
Volume	41
DOIs	https://doi.org/10.13001/ela.2025.9741
Publication status	Published - 1 Jan 2025

Bibliographical note

Publisher Copyright:
© 2025, International Linear Algebra Society. All rights reserved.

Keywords

Copositive matrix
Numerical range
Pareto eigenvalues

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10.13001/ela.2025.9741

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abstract = "We consider symmetric copositive matrices A ∈ Mn (R), which by definition satisfy xT Ax ≥ 0 for all nonzero x ≥ 0. We introduce the notion the copositive range of a copositive matrix A, CR(A) = \{xT Ax: x ≥ 0, ‖x‖2 = 1\}, and prove that CR(A) is an interval contained in the numerical range of A. We focus on the properties and the endpoints of CR(A), which are associated with the Pareto eigenvalues of A.",

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author = "Park, \{Seong Jun\} and Nurhan {\c C}olakoğlu and Tsatsomeros, \{Michael J.\}",

year = "2025",

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doi = "10.13001/ela.2025.9741",

language = "English",

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N2 - We consider symmetric copositive matrices A ∈ Mn (R), which by definition satisfy xT Ax ≥ 0 for all nonzero x ≥ 0. We introduce the notion the copositive range of a copositive matrix A, CR(A) = {xT Ax: x ≥ 0, ‖x‖2 = 1}, and prove that CR(A) is an interval contained in the numerical range of A. We focus on the properties and the endpoints of CR(A), which are associated with the Pareto eigenvalues of A.

AB - We consider symmetric copositive matrices A ∈ Mn (R), which by definition satisfy xT Ax ≥ 0 for all nonzero x ≥ 0. We introduce the notion the copositive range of a copositive matrix A, CR(A) = {xT Ax: x ≥ 0, ‖x‖2 = 1}, and prove that CR(A) is an interval contained in the numerical range of A. We focus on the properties and the endpoints of CR(A), which are associated with the Pareto eigenvalues of A.

KW - Copositive matrix

KW - Numerical range

KW - Pareto eigenvalues

UR - https://www.scopus.com/pages/publications/105019100325

U2 - 10.13001/ela.2025.9741

DO - 10.13001/ela.2025.9741

M3 - Article

AN - SCOPUS:105019100325

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VL - 41

SP - 575

EP - 587

JO - Electronic Journal of Linear Algebra

JF - Electronic Journal of Linear Algebra

ER -

THE COPOSITIVE RANGE

Abstract

Bibliographical note

Keywords

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