Star-Convexity of the Eigenvalue Regions for Stochastic Matrices and Certain Subclasses

Brando Vagenende (), Brecht Verbeken and Marie-Anne Guerry
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Brando Vagenende: Data Analytics Laboratory, Department of Business Technology and Operations, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium
Brecht Verbeken: Data Analytics Laboratory, Department of Business Technology and Operations, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium
Marie-Anne Guerry: Data Analytics Laboratory, Department of Business Technology and Operations, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium

Mathematics, 2025, vol. 13, issue 12, 1-10

Abstract: Star-convexity of the eigenvalue region for the set of n × n stochastic matrices has already been proven, for n ≥ 2 , by Dmitriev and Dynkin. The star-convexity property enables full determination of the eigenvalue region by its boundary. This study offers a more straightforward proof that extends to other subclasses of the stochastic matrices. Furthermore, the proof is constructive as it includes the explicit construction of the corresponding realizing matrices. Explicit sufficient conditions for star-convexity of the eigenvalue regions of stochastic subclasses are presented. In particular, star-convexity of the eigenvalue region is proved for the n × n doubly stochastic and the n × n monotone stochastic matrices.

Keywords: star-convexity; eigenvalues; eigenvalue regions; stochastic matrices; doubly stochastic matrices; monotone stochastic matrices (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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