SIGN PATTERNS THAT REQUIRE OR ALLOW POWER-POSITIVITY

This study investigates the conditions under which a sign pattern matrix is power-positive, which means that some positive integer power of the matrix is entrywise positive. The objective is to establish a clear criterion for determining when a sign pattern requires power-positivity. The methodology involves analyzing the relationships between the sign patterns of matrices A and -A, focusing on their nonnegative structures and their associated digraphs. Through rigorous analysis, it is shown that a sign pattern requires power-positivity if either the sign pattern A or its negation -A is nonnegative and exhibits a primitive digraph. Additionally, the study identifies the criteria for a sign pattern to be potentially power-positive, establishing that it is equivalent to the condition that A or -A is potentially eventually positive. The results provide a comprehensive understanding of the dynamics of power-positivity in matrices, highlighting its importance in the stability analysis of various systems, particularly in economic contexts. Furthermore, this research contributes valuable insights into the classification of sign patterns based on their power-positive properties.