# LINEAR OPERATORS THAT PRESERVE SETS OF PRIMITIVE MATRICES

• Beasley, Leroy B. (Department of Mathematics and Statistics Utah State University) ;
• Kang, Kyung-Tae (Department of Mathematics Jeju National University) ;
• Song, Seok-Zun (Department of Mathematics and Research Institute for Basic Sciences Jeju National University)
• Published : 2014.07.01

#### Abstract

We consider linear operators on square matrices over antinegative semirings. Let ${\varepsilon}_k$ denote the set of all primitive matrices of exponent k. We characterize those linear operators which preserve the set ${\varepsilon}_1$ and the set ${\varepsilon}_2$, and those that preserve the set ${\varepsilon}_{n^2-2n+2}$ and the set ${\varepsilon}_{n^2-2n+1}$. We also characterize those linear operators that strongly preserve ${\varepsilon}_2$, ${\varepsilon}_{n^2-2n+2}$ or ${\varepsilon}_{n^2-2n+1}$.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

#### References

1. R. Brualdi and H. Ryser, Combinatorial Matrix Theory, Cambridge University Press, New York, 1991.
2. L. B. Beasley and N. J. Pullman, Term-rank, permanent, and rook-polynomial preservers, Linear Algebra Appl. 90 (1987), 33-46. https://doi.org/10.1016/0024-3795(87)90302-8
3. L. B. Beasley and N. J. Pullman, Linear operators that strongly preserve primitivity, Linear and Multilinear Algebra 25 (1989), no. 3, 205-213. https://doi.org/10.1080/03081088908817942
4. L. B. Beasley and N. J. Pullman, Linear operators that strongly preserve the index of imprimitivity, Linear and Multilinear Algebra 31 (1992), no. 1-4, 267-283. https://doi.org/10.1080/03081089208818139
5. A. L. Dulmadge and N. S. Mendelsohn, The exponents of incident matrices, Duke Math. J. 31 (1964), 575-584. https://doi.org/10.1215/S0012-7094-64-03156-4
6. G. Frobenius, Uber die Darstellung der entlichen Gruppen durch Linear Substitutionen, S. B. Deutsch. Akad. Wiss. Berlin (1897), 994-1015.
7. J. C. Holladay and R. S. Varga, On powers of non-negative matrices, Proc. Amer. Math. Soc. 9 (1985), 631-634.
8. M. Lewin and Y. Vitek, A system of gaps in the exponent set of primitive matrices, Illinois J. Math. 25 (1981), no. 1, 87-98.
9. C.-K. Li and S. J. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), no. 7, 591-605. https://doi.org/10.2307/2695268
10. B. Liu, On the bounds of exponents of primitive (0, 1) matrices, Southeast Asia Bull. Math. 22 (1998), no. 1, 57-65.
11. S. J. Pierce, et al., A survey of linear preserver problems, Linear and Multilinear Algebra 33 (1992), no. 1-2, 1-129. https://doi.org/10.1080/03081089208818176
12. H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642-648. https://doi.org/10.1007/BF02230720
13. L. B. Beasley and A. E. Guterman, Operators preserving primitivity for matrix pairs, Matrix methods: theory, algorithms and applications, World Sci. Publ., Hackensack, NJ, 2-19, 2010.
14. L. B. Beasley and A. E. Guterman, The characterization of operators preserving primitivity for matrix k-tuples, Linear Algebra Appl. 430 (2009), no. 7, 1762-1777. https://doi.org/10.1016/j.laa.2008.06.031
15. L. B. Beasley and N. J. Pullman, Boolean rank preserving operators and Boolean rank-1 spaces, Linear Algebra Appl. 59 (1984), 55-77. https://doi.org/10.1016/0024-3795(84)90158-7