Conjectures about determining the regions of eigenvalues of stochastic and doubly stochastic matrices

Bara Kim; Jeongsim Kim

doi:10.1016/j.laa.2021.12.011

Conjectures about determining the regions of eigenvalues of stochastic and doubly stochastic matrices

Bara Kim, Jeongsim Kim

Department of Mathematics

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

1 Citation (Scopus)

Abstract

Let the regions Θ_n and ω_n be the subsets of the complex planes that consist of all eigenvalues of all n×n stochastic and doubly stochastic matrices, respectively. Also, let Π_n denote the convex hull of the nth roots of unity. Levick, Pereira and Kribs (2015) [10] made the following conjectures on the relations between Θ_n, ω_n and Π_n: ω_n=Θ_n−1∪Π_n and Θ_n−1⊂ω_n. These two conjectures are known to be true for n=2,3,4. In this paper, we will show that these two conjectures are not true for n≥5.

Original language	English
Pages (from-to)	157-174
Number of pages	18
Journal	Linear Algebra and Its Applications
Volume	637
DOIs	https://doi.org/10.1016/j.laa.2021.12.011
Publication status	Published - 2022 Mar 15

Bibliographical note

Funding Information:
We are grateful to the reviewer for their valuable comments. B. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIT ) (No. 2020R1A2B5B01001864 ). J. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIT ) (No. 2020R1F1A1A01065568 ).

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Doubly stochastic matrices
Eigenvalues
Stochastic matrices

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.2021.12.011

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title = "Conjectures about determining the regions of eigenvalues of stochastic and doubly stochastic matrices",

abstract = "Let the regions Θn and ωn be the subsets of the complex planes that consist of all eigenvalues of all n×n stochastic and doubly stochastic matrices, respectively. Also, let Πn denote the convex hull of the nth roots of unity. Levick, Pereira and Kribs (2015) [10] made the following conjectures on the relations between Θn, ωn and Πn: ωn=Θn−1∪Πn and Θn−1⊂ωn. These two conjectures are known to be true for n=2,3,4. In this paper, we will show that these two conjectures are not true for n≥5.",

keywords = "Doubly stochastic matrices, Eigenvalues, Stochastic matrices",

author = "Bara Kim and Jeongsim Kim",

note = "Funding Information: We are grateful to the reviewer for their valuable comments. B. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIT ) (No. 2020R1A2B5B01001864 ). J. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIT ) (No. 2020R1F1A1A01065568 ). Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

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doi = "10.1016/j.laa.2021.12.011",

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AU - Kim, Bara

AU - Kim, Jeongsim

N1 - Funding Information: We are grateful to the reviewer for their valuable comments. B. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIT ) (No. 2020R1A2B5B01001864 ). J. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIT ) (No. 2020R1F1A1A01065568 ). Publisher Copyright: © 2021 Elsevier Inc.

PY - 2022/3/15

Y1 - 2022/3/15

N2 - Let the regions Θn and ωn be the subsets of the complex planes that consist of all eigenvalues of all n×n stochastic and doubly stochastic matrices, respectively. Also, let Πn denote the convex hull of the nth roots of unity. Levick, Pereira and Kribs (2015) [10] made the following conjectures on the relations between Θn, ωn and Πn: ωn=Θn−1∪Πn and Θn−1⊂ωn. These two conjectures are known to be true for n=2,3,4. In this paper, we will show that these two conjectures are not true for n≥5.

AB - Let the regions Θn and ωn be the subsets of the complex planes that consist of all eigenvalues of all n×n stochastic and doubly stochastic matrices, respectively. Also, let Πn denote the convex hull of the nth roots of unity. Levick, Pereira and Kribs (2015) [10] made the following conjectures on the relations between Θn, ωn and Πn: ωn=Θn−1∪Πn and Θn−1⊂ωn. These two conjectures are known to be true for n=2,3,4. In this paper, we will show that these two conjectures are not true for n≥5.

KW - Doubly stochastic matrices

KW - Eigenvalues

KW - Stochastic matrices

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DO - 10.1016/j.laa.2021.12.011

M3 - Article

AN - SCOPUS:85121438620

SN - 0024-3795

VL - 637

SP - 157

EP - 174

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Conjectures about determining the regions of eigenvalues of stochastic and doubly stochastic matrices

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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