Polytropes and Tropical Eigenspaces: Cones of Linearity

The map which takes a square matrix A to its polytrope is piecewise linear. We show that cones of linearity of this map form a polytopal fan partition of , whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels (Bull. Lond. Math. Soc. 54:27-36, 2013) and lead to a new combinatorial classification of polytropes and tropical eigenspaces.