TY - JOUR

T1 - Conjectures about determining the regions of eigenvalues of stochastic and doubly stochastic matrices

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

UR - http://www.scopus.com/inward/record.url?scp=85121438620&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2021.12.011

DO - 10.1016/j.laa.2021.12.011

M3 - Article

AN - SCOPUS:85121438620

VL - 637

SP - 157

EP - 174

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -