  We study the tropicalization of the cone of positive semidefinite matrices
over the ordered field of real Puiseux series. The tropical PSD matrices form
the normal cone of the Newton polytope of the symmetric determinant at the
vertex corresponding to the product of diagonal entries. We find generators and
defining inequalities of the cone. The PSD tropical quadratic forms are those
that induce the trivial subdivision on the standard simplex dilated by two. We
also show that the tropical PSD cone is the tropical convex hull of the set of
symmetric matrices of tropical rank one and that every tropical PSD matrix can
be factored as a tropical product of a matrix and its transpose.
