An open set of 4×4 embeddable matrices whose principal logarithm is not a Markov generator

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 that are close enough to the identity matrix. In this paper we exhibit open sets of Markov matrices that are embeddable and whose principal logarithm is not a rate matrix, thus proving that the principal logarithm test above does not suffice generically. ; All authors are partially funded by AGAUR Project 2017 SGR-932 and MINECO/FEDER Projects MTM2015-69135 and MDM-2014-0445. J. Roca-Lacostena has received also funding from Secretaria d'Universitats i Recerca de la Generalitat de Catalunya (AGAUR 2018FI_B_00947) and European Social Funds. ; Postprint (author's final draft)