Minimal polynomials, scaled Jordan frames, and Schur-type majorization in hyperbolic systems

Abstract

Corresponding to a hyperbolic system $(V, p, e)$, where $V$ is a real finite-dimensional vector space and $p$ is a hyperbolic polynomial of degree $n$ in the direction $e$, we consider the eigenvalue map $\lambda: V \to R^n$ and the hyperbolicity cone $\Lambda_+$. In such a system, a scaled Jordan frame is defined as a finite set of rank-one elements whose sum lies in the interior of $\Lambda_+$. We show that when the system has a scaled Jordan frame and $n \geq 2$, $p$ and its derivative polynomial $p^\prime$ are minimal polynomials (generating their respective hyperbolicity cones), thereby extending a result of Ito and Louren{\c c}o proved in the setting of a rank-one generated (proper) hyperbolicity cone. When each element of a scaled Jordan frame has trace one and the total sum is $e$ (such a set is called a Jordan frame), we show that the frame is orthonormal relative to the semi-inner product induced by $\lambda$ with exactly $n$ elements, and $V$ contains a copy of $R^n$ (as a Euclidean Jordan algebra). We also present a Schur-type majorization result corresponding to a Jordan frame and an $e$-doubly stochastic $n$-tuple.