On some sign patterns of algebraically positive matrices

The concept of an algebraically positive matrix was introduced by Kirkland, Qiao, and Zhan in 2016. A real square matrix A is said to be algebraically positive if there exists a real polynomial f such that f(A) is a positive matrix. In this paper, we give a new characterization of algebraically positive matrices, and characterize all tree sign pattern matrices that allow algebraic positivity, and all star and path sign pattern matrices that require algebraic positivity. Also, all tree sign pattern matrices of order less than 6 requiring algebraic positivity are characterized.