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A Markov matrix is embeddable if it can represent a homogeneouscontinuous-time Markov process. It is well known that if a Markov matrix hasreal and pairwise-different eigenvalues, then the embeddability can bedetermined 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 thatrule a Markov process subjected to certain restrictions. In this paper we provethat this criterion cannot be generalized and we provide open sets of Markovmatrices that are embeddable and whose principal logarithm is a not a ratematrix.

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