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Tree Sign Patterns of Small Order that Allow an Eventually Positive Matrix

Authors: Ber-Lin Yu, Jie Cui, Hong Cheng, Zhengfeng Yu

Abstract:

A sign pattern is a matrix whose entries belong to the set {+,−, 0}. An n-by-n sign pattern A is said to allow an eventually positive matrix if there exist some real matrices A with the same sign pattern as A and a positive integer k0 such that Ak > 0 for all k ≥ k0. It is well known that identifying and classifying the n-by-n sign patterns that allow an eventually positive matrix are posed as two open problems. In this article, the tree sign patterns of small order that allow an eventually positive matrix are classified completely.

Keywords: Eventually positive matrix, sign pattern, tree.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1338578

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References:


[1] R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge: Cambridge University Press, 1991.
[2] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 1995.
[3] A. Berman, M. Catral, L. M. Dealba, A. Elhashash, F. Hall, L. Hogben, I. J. Kim, D. D. Olesky, P. Tarazaga, M. J. Tsatsomeros, P. van den Driessche, Sign patterns that allow eventual positivity, Electronic Journal of Linear Algebra 19:108-120, 2010.
[4] B. -L. Yu, T. -Z. Huang, J. Luo, H. B. Hua, Potentially eventually positive double star sign patterns, Applied Mathematics Letters, 25:1619-1624, 2012.
[5] B. -L. Yu, T. -Z. Huang, On minimal potentially power-positive sign patterns, Operators and Matrices, 6:159-167, 2012.
[6] B. -L. Yu, T. -Z. Huang, C. Hong, D. D. Wang, Eventual positivity of tridiagonal sign patterns, Linear and Multilinear Algebra, 62:853-859, 2014.
[7] M. Archer, M. Catral, C. Erickson, R. Haber, L. Hogben, X. Martinez-Rivera, A. Ochoa, Constructions of potentially eventually positive sign patterns with reducible positive part, Involve, 4:405-410, 2011.
[8] E. M. Ellison, L. Hogben, M. J. Tsatsomeros, Sign patterns that require eventual positivity or require eventual nonnegativity, Electronic Journal of Linear Algebra, 19:98-107, 2010.