Polynomials that preserve nonnegative matrices of order two

TL;DR

This paper characterizes polynomials that preserve nonnegative matrices of order two, introduces new conditions, and establishes inclusion relations among polynomial classes, with implications for circulant matrices.

Contribution

It provides a new characterization of polynomials preserving nonnegative matrices of order two and proves the inclusion _3 \u2286 _2, expanding understanding of matrix-preserving polynomials.

Findings

Characterization of polynomials preserving nonnegative matrices of order two

Proof that _3 is a subset of _2

New characterization for polynomials preserving nonnegative circulant matrices of order two

Abstract

A known characterization for entire functions that preserve all nonnegative matrices of order two is shown to characterize polynomials that preserve nonnegative matrices of order two. Equivalent conditions are derived and used to prove that $\mathscr{P}_3 \subset \mathscr{P}_2$, which was previously unknown. A new characterization is given for polynomials that preserve nonnegative circulant matrices of order two.