Tree Sign Patterns of Small Order that Allow an Eventually Positive Matrix
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71663", author = "Ber-Lin Yu and Jie Cui and Hong Cheng and Zhengfeng Yu", title = "Tree Sign Patterns of Small Order that Allow an Eventually Positive Matrix", 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.", volume = "9", number = "4", pages = "252-4", }