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

TL;DR

This paper characterizes specific stochastic matrices related to the boundary of the Karpelevi
0c region, providing explicit structures for matrices with certain polynomial types and insights into their extreme eigenvalues.

Contribution

It explicitly characterizes all stochastic matrices with Type 0 or I polynomials and the sparsest matrices with Type II or III polynomials, advancing understanding of their boundary eigenvalues.

Findings

Explicit characterization of matrices with Type 0 and I polynomials.

Identification of sparsest matrices with Type II and III polynomials.

Deeper insights into the structure of matrices with extreme eigenvalues.

Abstract

A celebrated result of Karpelevi\v c describes $\Theta_n,$ the collection of all eigenvalues arising from the stochastic matrices of order $n.$ The boundary of $\Theta_n$ 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 $n \times n$ 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.