Convex-Cyclic Matrices, Convex-Polynomial Interpolation & Invariant Convex Sets

TL;DR

This paper investigates convex-polynomials and their relation to matrix dynamics, characterizing convex-cyclic matrices and their invariant convex sets, providing new proofs and insights into matrix invariance properties.

Contribution

It introduces a new characterization of convex-cyclic matrices and establishes connections between convex-polynomials, matrix invariance, and interpolation, correcting previous results.

Findings

Characterization of convex-cyclic matrices

Prescribing values and derivatives of convex-polynomials

Invariant convex sets correspond to invariant subspaces

Abstract

We define a convex-polynomial to be one that is a convex combination of the monomials $\{1, z, z^2, \ldots\}$. This paper explores the intimate connection between peaking convex-polynomials, interpolating convex-polynomials, invariant convex sets, and the dynamics of matrices. In particular, we use these intertwined relations to both prove which matrices are convex-cyclic while at the same time proving that we can prescribe the values and a finite number of the derivatives of a convex-polynomial subject to certain natural constraints. These properties are also equivalent to determining those matrices whose invariant closed convex sets are all invariant subspaces. Our characterization of the convex-cyclic matrices gives a new and correct proof of a similar result by Rezaei that was stated and proven incorrectly.