Proofs of conjectures on the Karpelevich arcs in the region of eigenvalues of stochastic matrices

Bara Kim; Jeongsim Kim

doi:10.1016/j.laa.2020.02.029

Proofs of conjectures on the Karpelevich arcs in the region of eigenvalues of stochastic matrices

Bara Kim, Jeongsim Kim

Department of Mathematics

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

2 Citations (Scopus)

Abstract

The subset of the complex plane that consists of all eigenvalues of all stochastic matrices of a fixed order was completely determined by Karpelevich (1951). The boundary of this region consists of so-called Karpelevich arcs. Johnson and Paparella (2017) [6] made several conjectures on properties of these arcs. In this paper, we prove two of their conjectures. Specifically, we prove that the Karpelevich arcs are regular differentiable curves and establish that some powers of certain Karpelevich arcs correspond to some other Karpelevich arcs.

Original language	English
Pages (from-to)	13-23
Number of pages	11
Journal	Linear Algebra and Its Applications
Volume	595
DOIs	https://doi.org/10.1016/j.laa.2020.02.029
Publication status	Published - 2020 Jun 15

Bibliographical note

Funding Information:
We are grateful to the reviewer for valuable comments and suggestions, which greatly improved this paper. B. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B01001864 ).

Publisher Copyright:
© 2020 Elsevier Inc.

Keywords

Farey pair
Karpelevich arc
Stochastic matrix

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.2020.02.029

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abstract = "The subset of the complex plane that consists of all eigenvalues of all stochastic matrices of a fixed order was completely determined by Karpelevich (1951). The boundary of this region consists of so-called Karpelevich arcs. Johnson and Paparella (2017) [6] made several conjectures on properties of these arcs. In this paper, we prove two of their conjectures. Specifically, we prove that the Karpelevich arcs are regular differentiable curves and establish that some powers of certain Karpelevich arcs correspond to some other Karpelevich arcs.",

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AB - The subset of the complex plane that consists of all eigenvalues of all stochastic matrices of a fixed order was completely determined by Karpelevich (1951). The boundary of this region consists of so-called Karpelevich arcs. Johnson and Paparella (2017) [6] made several conjectures on properties of these arcs. In this paper, we prove two of their conjectures. Specifically, we prove that the Karpelevich arcs are regular differentiable curves and establish that some powers of certain Karpelevich arcs correspond to some other Karpelevich arcs.

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ER -

Proofs of conjectures on the Karpelevich arcs in the region of eigenvalues of stochastic matrices

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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