Characterizations of Embeddable 3 x 3 Stochastic Matrices with a Negative Eigenvalue

The problem of identifying a stochastic matrix as a transition matrix between two xed times say t and t of a continuous time and nite state Markov chain has been shown to have practical importance especially in the area of stochastic models applied to social phenomena The embedding problem of nite Markov chains as it is called comes down to investigating whether the stochastic matrix can be expressed as the exponential of some matrix with row sums equal to zero and nonnegative o diagonal elements The aim of this paper is to answer a question left open by S Johansen i e to characterize those stochastic matrices of order three with an eigenvalue of multiplicity