Spectral linear matrix inequalities
arXiv:2008.13452 · doi:10.1016/j.aim.2021.107749
Abstract
We prove, under a certain representation theoretic assumption, that the set of real symmetric matrices, whose eigenvalues satisfy a linear matrix inequality, is itself a spectrahedron. The main application is that derivative relaxations of the positive semidefinite cone are spectrahedra. From this we further deduce statements on their Wronskians. These imply that Newton's inequalities, as well as a strengthening of the correlation inequalities for hyperbolic polynomials, can be expressed as sums of squares.
References in corpus (11)
- Polynomials with the half-plane property and matroid theory
- Homogeneous multivariate polynomials with the half-plane property
- Obstructions to determinantal representability
- Hyperbolic polynomials, interlacers, and sums of squares
- Determinantal Representations and Bézoutians
- Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones
- Two results on the size of spectrahedral descriptions
- A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone
- Spectrahedral representations of plane hyperbolic curves
- Exponential lower bounds on spectrahedral representations of hyperbolicity cones
- Highly symmetric matroids, the strong Rayleigh property, and sums of squares