Sign patterns that allow eventual positivity

Date
2010-02-01
Authors
Berman, Abraham
Catral, Minerva
DeAlba, Luz
Elhashash, Abed
Hall, Frank
Hogben, Leslie
Kim, In-Jae
Olesky, Dale
Tarazaga, Pablo
Tsatsomeros, Michael
van den Driessche, Pauline
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Abstract

Several necessary or sufficient conditions for a sign pattern to allow eventual positivity are established. It is also shown that certain families of sign patterns do not allow eventual positivity. These results are applied to show that for n ≥ 2, the minimum number of positive entries in an n×n sign pattern that allows eventual positivity is n+1, and to classify all 2×2 and 3×3 sign patterns as to whether or not the pattern allows eventual positivity. A 3 × 3 matrix is presented to demonstrate that the positive part of an eventually positive matrix need not be primitive, answering negatively a question of Johnson and Tarazaga.

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This article is published as Berman, Abraham, Minerva Catral, Luz DeAlba, Abed Elhashash, Frank Hall, Leslie Hogben, In-Jae Kim et al. "Sign patterns that allow eventual positivity." The Electronic Journal of Linear Algebra 19 (2009): 108-120. DOI: 10.13001/1081-3810.1351 . Posted with permission.

Keywords
Eventually positive matrix, Potentially eventually positive sign pattern, Perron-Frobenius, Directed graph
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