On $e$-doubly stochastic matrices over hyperbolic (polynomial) systems
In the setting of a hyperbolic (polynomial) system $(\mathcal{V}, p, e)$ of degree $n$, an $n$-tuple $\mathbf{A} = \big[ a_1, a_2, \dots, a_n \big]$ is said to be an $e$-doubly stochastic $\mathcal{V}$-matrix if each $a_i$ belongs to the hyperbolicity cone, has trace $1$, and $a_1 + a_2 + \dots + a_n = e$. In this article, we characterize linear preservers of such $\mathcal{V}$-matrices, describe some connections between $e$-doubly stochasticity and majorization, and study extreme points of the set of all $e$-doubly stochastic $\mathcal{V}$-matrices. We show, for example, that $(i)$ when $n>1$, positive, unital, and trace-preserving transformations are (the only) linear transformations on $\mathcal{V}$ that preserve $e$-doubly stochasticity; $(ii)$ when $\mathbf{A}$ is $e$-doubly stochastic, the eigenvalue vector of the linear combination $\sum_{i=1}^{n} r_ia_i$ is majorized by the coefficient vector $(r_1, r_2, \dots, r_n)^{T} \in \mathbb{R}^n$; and $(iii)$ when $p$ is complete, $e$-doubly stochastic $\mathcal{V}$-matrices induced by (generalized) Jordan frames are extreme points of the compact convex set of all $e$-doubly stochastic $\mathcal{V}$-matrices.