Abstract and real matrix structures for hyperbolicity cones

Abstract

Hyperbolic polynomials are a certain class of multi-variate polynomials that have all real roots (eigenvalues) in a specified direction. Likewise hyperbolicity cones are a certain class of cones arising from hyperbolic polynomials which maintain some of the important properties of positive semi-definite cones (PSD) including, but not limited to, convexity.We first present a representation of hyperbolicity cones in terms of abstract PSD cones over a space of super-symmetric abstract matrices. In the process we also discover a new perspective on some classic identities of Isaac Newton. Next, we show two ways in which the above result can be expressed in terms of real matrices.In the last chapter we return to our abstract matrices and examine the three by three case. Special connections to notion of a self-concordant barrier functions on arbitrary convex cones arise.The appendix introduces our Matlab Hyperbolic Polynomial Toolbox.