Spectral linear matrix inequalities

TL;DR

This paper proves that certain eigenvalue conditions define spectrahedra under specific assumptions, with applications to derivative relaxations of cones and implications for inequalities in hyperbolic polynomials.

Contribution

It establishes that eigenvalue-based linear matrix inequalities form spectrahedra under a representation theoretic assumption, extending the understanding of spectrahedral sets.

Findings

Derivative relaxations of the positive semidefinite cone are spectrahedra.

Newton's inequalities can be expressed as sums of squares.

Results connect eigenvalue inequalities with spectrahedral representations.

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.