We introduce the study of potentially eventually exponentially positive (PEEP) sign patterns and establish several results using the connections between these sign patterns and the potentially eventually positive (PEP) sign patterns. It is shown that the problem of characterizing PEEP sign patterns is not equivalent to that of characterizing PEP sign patterns. A characterization of all 2 × 2 and 3 × 3 PEEP sign patterns is given.
Potentially eventually positive (PEP) sign patterns were introduced by Berman et al. (Electron. J. Linear Algebra 19 (2010), 108-120), where it was noted that a matrix is PEP if its positive part is primitive, and an example was given of a 3 × 3 PEP sign pattern with reducible positive part. We extend these results by constructing n×n PEP sign patterns with reducible positive part, for every n ≥ 3.
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