Abstract
Hyperbolic polynomials are multivariate homogeneous polynomials with certain real-rootedness properties. One family of examples is the determinant restricted to certain subspaces of symmetric matrices. Associated with any hyperbolic polynomial is a convex `hyperbolicity' cone. Any spectrahedron (linear `slice' of a positive semidefinite cone) is an example of a hyperbolicity cone. The generalized Lax conjecture asks whether, conversely, every hyperbolicity cone is a spectrahedron.
One important way to construct new hyperbolic polynomials is by taking directional derivatives of known hyperbolic polynomials. We show that the
hyperbolicity cone associated with the directional derivative of the determinant in the identity direction is a spectrahedron. This gives rise to a new family of explicit examples for which the generalized Lax conjecture holds.