Sign patterns that allow algebraic positivity

TL;DR

This paper explores conditions under which sign pattern matrices permit algebraic positivity, providing methods for constructing such matrices and proposing conjectures to advance understanding in this area.

Contribution

It offers a sufficient condition for sign pattern matrices to allow algebraic positivity and introduces methods to build higher-order algebraically positive matrices from lower-order ones.

Findings

Provided a sufficient condition for sign pattern matrices to allow algebraic positivity.

Developed methods to construct higher-order algebraically positive matrices.

Proposed two conjectures related to algebraic positivity in sign pattern matrices.

Abstract

A real square matrix is algebraically positive if there exists a real polynomial $f$ such that $f(A)$ is a positive matrix. In this paper, we give a sufficient condition for a sign pattern matrix to allow algebraic positivity, and give some methods to construct higher-order algebraically positive matrices from some lower-order algebraically positive matrices. We also propose two conjectures related to the problem of allowing algebraic positivity.