  If a real symmetric matrix of linear forms is positive definite at some
point, then its determinant is a hyperbolic hypersurface. In 2007, Helton and
Vinnikov proved a converse in three variables, namely that every hyperbolic
plane curve has a definite real symmetric determinantal representation. The
goal of this paper is to give a more concrete proof of a slightly weaker
statement. Here we show that every hyperbolic plane curve has a definite
determinantal representation with Hermitian matrices. We do this by relating
the definiteness of a matrix to the real topology of its minors and extending a
construction of Dixon from 1902. Like Helton and Vinnikov's theorem, this
implies that every hyperbolic region in the plane is defined by a linear matrix
inequality.

Algebraic Geometry;Optimization and Control