Embedding of reversible Markov matrices

TL;DR

This paper revisits the conditions under which reversible Markov matrices can be embedded into continuous-time Markov processes, exploring relaxations like non-irreducibility and negative eigenvalues to broaden understanding.

Contribution

It extends previous work by relaxing assumptions such as irreducibility and considering matrices with negative eigenvalues, offering new insights into embeddability conditions.

Findings

Reversible Markov matrices can be embedded without requiring irreducibility.

Weakly reversible matrices are also considered for embeddability.

Negative eigenvalues in reversible matrices are analyzed for embeddability.

Abstract

The embeddability of reversible Markov matrices into time-homogeneous Markov semigroups is revisited, with some focus on simplifications and extensions. In particular, we do not demand irreducibility and consider weakly reversible matrices as well as reversible matrices with negative eigenvalues.