On properties of eigenvalue regions for monotone stochastic matrices

Monotone stochastic matrices are stochastic matrices in which each row stochastically dominates the previous one. While the eigenvalue regions for stochastic matrices have been fully described by F.I. Karpelevich in 1951, this study focuses on the analysis of monotone matrices. This paper examines their spectral properties and establishes a reduction theorem stating that, for n from 3 on, the eigenvalue region for the n x n monotone matrices is included in those for the (n-1) x (n-1) stochastic matrices. Moreover, the eigenvalue region, along with the corresponding realising matrices, is determined for monotone matrices up till order 3.