On equal-input and monotone Markov matrices

TL;DR

This paper investigates the properties and relationships of equal-input and monotone Markov matrices, focusing on embeddability, infinite divisibility, and algebraic structures, providing new uniqueness results for the Markov embedding problem.

Contribution

It offers new insights into the structure of these matrices, including several uniqueness theorems, using algebraic and geometric methods.

Findings

Characterization of embeddability conditions

New uniqueness results for Markov embedding

Analysis of idempotent Markov matrices

Abstract

The practically important classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the classic Markov embedding problem are obtained in the process. To achieve our results, we need to employ various algebraic and geometric tools, including commutativity, permutation invariance and convexity. Of particular relevance in several demarcation results are Markov matrices that are idempotents.