Stochastic Matrices Realising the Boundary of the Karpelevi\v c Region

2021/10/03 by Stephen Kirkland, Kirkland, Stephen, Helena Šmigoc +1 · 1 citation
Mathematics · Physics and Astronomy · #15A18 #15B51 #60J10 #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Probability (math.PR) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2110.01040

openalex publication_date 2021/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A celebrated result of Karpelevi\v c describes Θ_n, the collection of all eigenvalues arising from the stochastic matrices of order n. The boundary of Θ_n consists of roots of certain one-parameter families of polynomials, and those polynomials are naturally associated with the so--called reduced Ito polynomials of Types 0, I, II and III. In this paper we explicitly characterise all n × n stochastic matrices whose characteristic polynomials are of Type 0 or Type I, and all sparsest stochastic matrices of order n whose characteristic polynomials are of Type II or Type III. The results provide insights into the structure of stochastic matrices having extreme eigenvalues.

Cited by

• The Type III realisation conjecture of Kirkland and Šmigoc

Related

• The Karpelevič Region Revisited
• Demystifying the Karpelevic theorem
• Powers of Karpelevic arcs and their Sparsest Realising matrices
• Eigenvalue regions and realising monotone stochastic matrices
• The Type III realisation conjecture of Kirkland and Šmigoc