On Sign Patterns of 4 × 4 Algebraically Positive Matrices

A square matrix A with real entries is said to be algebraically positive (AP) if there exists a real polynomial f such that fA is a positive matrix. A square sign pattern matrix A is said to allow algebraic positivity if there is an AP matrix A whose sign pattern class is A, and A is said to require algebraic positivity if every matrix A having sign pattern class A is AP. In this paper, we consider 4×4 sign pattern matrices and investigate which of them allow algebraic positivity or require algebraic positivity. By studying 4×4 symmetric zero-diagonal sign pattern matrices, we list all nonequivalent 4×4 symmetric zero-diagonal sign pattern matrices and give some characterizations of 4×4 symmetric zero-diagonal sign pattern matrices that do not allow algebraic positivity, allow but do not require algebraic positivity, or require algebraic positivity. We also determine a class of 4×4 symmetric sign pattern matrices that allow algebraic positivity.