  A reciprocal linear space is the image of a linear space under
coordinate-wise inversion. These fundamental varieties describe the analytic
centers of hyperplane arrangements and appear as part of the defining equations
of the central path of a linear program. Their structure is controlled by an
underlying matroid. This provides a large family of hyperbolic varieties,
recently introduced by Shamovich and Vinnikov. Here we give a definite
determinantal representation to the Chow form of a reciprocal linear space. One
consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal
linear spaces. Another is a representation of the entropic discriminant as a
sum of squares. For generic linear spaces, the determinantal formulas obtained
are closely related to the Laplacian of the complete graph and generalizations
to simplicial matroids. This raises interesting questions about the
combinatorics of hyperbolic varieties and connections with the positive
Grassmannian.
