An open set of $4\times4$ embeddable matrices whose principal logarithm is not a Markov generator

TL;DR

This paper demonstrates that the common criterion for embeddability of Markov matrices, based on the principal logarithm being a rate matrix, does not apply universally, by providing open sets of matrices where this criterion fails.

Contribution

The authors prove that the principal logarithm criterion for embeddability cannot be generalized and identify open sets of matrices where it does not hold.

Findings

Principal logarithm is not always a rate matrix for embeddable Markov matrices.

Open sets of embeddable matrices exist with non-rate principal logarithms.

The criterion for embeddability based on the principal logarithm is not universally valid.

Abstract

A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the embeddability can be determined by checking whether its principal logarithm is a rate matrix or not. The same holds for Markov matrices close enough to the identity matrix or that rule a Markov process subjected to certain restrictions. In this paper we prove that this criterion cannot be generalized and we provide open sets of Markov matrices that are embeddable and whose principal logarithm is a not a rate matrix.