Conjectures about determining the regions of eigenvalues of stochastic and doubly stochastic matrices

Abstract

Let the regions circle minus(n) and omega(n) be the subsets of the complex planes that consist of all eigenvalues of all n x n stochastic and doubly stochastic matrices, respectively. Also, let Pi(n) denote the convex hull of the nth roots of unity. Levick, Pereira and Kribs (2015) [10] made the following conjectures on the relations between circle minus(n), omega(n) and Pi(n): omega(n) = circle minus(n-1) boolean OR Pi(n) and circle minus(n-1) subset of omega(n). These two conjectures are known to be true for n = 2, 3, 4. In this paper, we will show that these two conjectures are not true for n >= 5. (c) 2021 Elsevier Inc. All rights reserved.