  We give a complete description of Green's D relation for the multiplicative
semigroup of all n-by-n tropical matrices. Our main tool is a new variant on
the duality between the row and column space of a tropical matrix (studied by
Cohen, Gaubert and Quadrat and separately by Develin and Sturmfels). Unlike the
existing duality theorems, our version admits a converse, and hence gives a
necessary and sufficient condition for two tropical convex sets to be the row
and column space of a matrix. We also show that the matrix duality map induces
an isometry (with respect to the Hilbert projective metric) between the
projective row space and projective column space of any tropical matrix, and
establish some foundational results about Green's other relations.
