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

TL;DR

This paper explores the structure of hyperbolic systems, establishing conditions under which polynomials are minimal and characterizing Jordan frames, with implications for Schur-type majorization in hyperbolic cones.

Contribution

It extends known results about minimal polynomials and Jordan frames in hyperbolic systems, linking scaled Jordan frames to Euclidean Jordan algebras and majorization theory.

Findings

When a scaled Jordan frame exists, $p$ and $p'$ are minimal polynomials.

Jordan frames with trace-one elements form orthonormal bases in $V$.

A Schur-type majorization result is established for 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.