ON MINIMAL POTENTIALLY POWER-POSITIVE SIGN PATTERNS

An n-by-n sign pattern A is said to be potentially power-positive if there exists some A is an element of Q(A) such that A is power-positive, i.e., A(k) > 0 for some positive integer k. Catral, Hogben, Olesky and van den Driessche [Sign patterns that require or allow power-positivity, Electron. J. Linear Algebra, 19 (2010), 121-128] investigated the sign patterns that require or allow power-positivity. It has been shown that an n-by-n sign pattern A is potentially power-positive if and only if either A or -A is potentially eventually positive. But as the identification of sufficient and necessary conditions for potentially eventually positive sign patterns remains open, the characterization of potentially power-positive sign patterns is still open. In this paper, we introduce the minimal potentially power-positive sign patterns to classify the potentially power-positive sign patterns. Some properties of minimal potentially power-positive sign patterns are presented. It is shown that for an n-by-n sign pattern A with at most n+1 negative entries, A is minimal potentially power-positive if and only if either A or -A is minimal potentially eventually positive. Finally, we classify the minimal potentially power-positive sign patterns of order n <= 3.