Spectral linear matrix inequalities

We prove, under a certain representation theoretic assumption, that the set of real symmetric matrices, whose eigenvalues satisfy linear matrix inequality, is itself spectrahedron. The main application derivative relaxations positive semidefinite cone are spectrahedra. From this we further deduce statements on their Wronskians. These imply Newton's inequalities, as well strengthening correlation inequalities for hyperbolic polynomials, can be expressed sums squares.