The Karpelevič Region Revisited

2020/05/05 by Stephen Kirkland, Kirkland, Stephen, Thomas J. Laffey +3
Arts and Humanities · Computer Science · Engineering · Mathematics · #15A18 #15B51 #60J10 #Advanced Combinatorial Mathematics #FOS: Mathematics #Linguistics and language evolution #Matrix Theory and Algorithms #Spectral Theory (math.SP) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2005.02452

openalex publication_date 2020/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Karpelevič region Θ_n ⊂ ℂ consisting of all eigenvalues of all stochastic matrices of order n. We provide an alternative characterisation of Θ_n that sharpens the original description given by Karpelevič. In particular, for each θ∈ [0, 2π), we identify the point on the boundary of Θ_n with argument θ. We further prove that if n ∈ ℕ with n ≥ 2, and t ∈ Θ_n, then t is a subdominant eigenvalue of some stochastic matrix of order n.

Related

• Demystifying the Karpelevic theorem
• Stochastic Matrices Realising the Boundary of the Karpelevi\v c Region
• Powers of Karpelevic arcs and their Sparsest Realising matrices
• Eigenvalue regions and realising monotone stochastic matrices
• On the eigenvalue region of permutative doubly stochastic matrices