Eigenvalue regions and realising monotone stochastic matrices

TL;DR

This paper explores the spectral properties of monotone stochastic matrices, characterizing their eigenvalue regions and providing conditions for their realization, revealing smaller eigenvalue regions compared to general stochastic matrices.

Contribution

It introduces a detailed analysis of eigenvalue regions for monotone stochastic matrices, including complete determination for sizes up to 3 and a reduction theorem for larger sizes.

Findings

Eigenvalue regions for monotone matrices up to order 3 are fully characterized.

Pairs of non-trivial eigenvalues for 3x3 monotone matrices are characterized with realising matrices.

Eigenvalue regions for n x n monotone matrices are contained within those of (n-1) x (n-1) stochastic matrices for n ≥ 4.

Abstract

Eigenvalues of stochastic matrices have been studied from two complementary perspectives. The individual eigenvalues are characterised through the well-established Karpelevich regions. The spectrum as a whole has also been analysed, yielding powerful results such as the Johnson-Loewy-London (JLL) inequalities. Current research now turns toward particular subsets of stochastic matrices, among others the doubly stochastic matrices. This paper studies spectral properties of monotone stochastic matrices which are characterised by the fact that each row stochastically dominates the preceding one, and which arise in contexts such as intergenerational mobility, equal-input models, and credit-rating systems. This paper analyses the dominance matrix associated with a monotone matrix, which is a non-negative matrix that preserves the non-trivial eigenvalues. Properties are established and the conditions are given under which a non-negative matrix can be regarded as a dominance matrix. In analogy with the stochastic matrices, this study examines for the monotone stochastic matrices both the individual eigenvalues as the spectrum as a whole. Individually, the eigenvalue region for all monotone matrices up till order 3 is completely determined, and realising matrices are provided. Collectively, the set of possible pairs of non-trivial eigenvalues arising from 3x3 monotone matrices is characterised, accompanied by realising matrices. In both perspectives, the resulting regions are substantially smaller than those for general stochastic matrices. Finally, this paper proves a reduction theorem stating that, for all n from 4 on, the eigenvalue region of n x n monotone matrices is contained within that of (n-1) x (n-1) stochastic matrices.