Abstract
We give a counterexample to the Strong Bang-Bang Conjecture according to which any 3 × 3 embeddable matrix can be expressed as a product of six Poisson matrices. We exhibit a 3 × 3 embeddable matrix which can be expressed as a product of seven but not six Poisson matrices. We show that an embeddable 3 × 3 matrix P with det P ≥ 1 8 can be expressed as a product of at most six Poisson matrices and give necessary and sufficient conditions for a 3 × 3 stochastic matrix P with det P ≥ 1 8 to be embeddable. For an embeddable 3 × 3 matrix P with det P < 1 8 we give a new bound for the number of Poisson matrices in its Bang-Bang representation.
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