# Notes on Markov embedding

## TL;DR

This paper revisits the representation problem of finite-dimensional Markov matrices in Markov semigroups, focusing on algebraic properties and criteria for specific subclasses like circulant and doubly stochastic matrices.

## Contribution

It provides new criteria and insights into the embedding problem for various subclasses of Markov matrices, emphasizing algebraic structures and the centraliser connection.

## Key findings

- Criteria for embedding of specific matrix subclasses
- Analysis of algebraic properties related to the embedding problem
- Connection between Markov matrices and their centralisers

## Abstract

The representation problem of finite-dimensional Markov matrices in Markov semigroups is revisited, with emphasis on concrete criteria for matrix subclasses of theoretical or practical relevance, such as equal-input, circulant, symmetric or doubly stochastic matrices. Here, we pay special attention to various algebraic properties of the embedding problem, and discuss the connection with the centraliser of a Markov matrix.