The structure of strongly linear preservers of degree majorization

Document Type : Original Paper

Authors

Department of Mathematics, Sirjan University of Technology, Sirjan, Iran

Abstract

A square matrix $D$ is ​​called a ‎doubly‎ stochastic matrix if all its entries are non-negative and the sum of the entries of each row is equal to the sum of the entries of each column and is equal to one. For each linear and non-zero vector $x = (x_1, ‎‎\ldots, x_n)$, we define the degree of $x$ as the largest number ‎$‎i‎$‎ such that $x_i$ is non-zero and the degree of vector zero is zero. We say that a vector $x$ is degree majorized by $y$ and denote by $x\prec_{deg} y$ if the degree of $x$ is greater than or equal to the degree of $y$ and $x = yD$ for some doubly stochastic matrix $D$. ‎

In this paper, we ‎obtain‎ the structure of all linear preservers of degree majorization on space ‎‎$‎‎\mathbb{R}^2‎$‎‎‏‎. Also, we find the structure of all strong linear preservers of degree majorization on real vector spaces $‎‎\mathbb{R}^‎n‎‎$.

Keywords

Main Subjects

References

[1] ع. آرمندنژاد، مروری بر مهتری های عادی و تعمیم یافته و بررسی ساختار نگه دارنده های خطی آن ها، فرهنگ و اندیشه ریاضی، ۴۵، (1389)، 31-40.
[2] ا. محمدحسنی، ی. سیاری، ماتریس های زیر تصادفی سطری تعمیم یافته و مهتری، مدل سازی پیشرفته ریاضی، 12 (4)  (1401)، 523-534.
[3]  ا. محمدحسنی، ی. سیاری، م. سبزواری، نگه دارنده های خطی مهتر راست-چپ ماتریسی، موجک ها و جبرخطی، 8 (3) (1401)، 37-59.
[4] A. Armandnejad and Z. Gashool, Strong linear preservers of g-tridiagonal majorization on Rn, Elec. J. Linear Algeb. 123 (2012), 115–121.
[5] A. Armandnejad and A. Salemi, The structure of linear preservers of gs-majorization, Bull. Iranian Math. Soc. 32 (2) (2006), 31–42.
[6] R. Bahatia, Matrix Analysis, Springer-Verlag, New York, 1997.
[7] L. B. Beasley, S.G. Lee and Y.H. Lee, A characterization of strong preservers of matrix majorization, Linear Algebra Appl. 367 (2003), 341–346.
[8] R. A. Brualdi and G. Dahl, An extension of the polytope of doubly stochastic matrices, Linear and Multilinear Algebra, 6(3) (2013), 393–408.
[9] G.S. Cheon and Y.H. Lee, The doubly stochastic matrices of a multivariate majorization, J. Kor. Math. Soc. 32 (1995), 857–867.
[10] H. Chiang and C.K. Li, Generalized doubly stochastic matrices and linear preservers, Linear and Multilinear Algebra, 53 (2005), 1–11.
[11] G. Dahl, Matrix majorization, Linear Algebra Appl. 288 (1999), 53–73.
[12] M. Dehghanian and A. Mohammadhasani, A note on multivariate majorization, J. Mahani Math. Res. Cent. 11(2) (2022), 119–126.
[13] M.H. Hadian and A. Armandnejad, B-majorization and its linear preservers, Linear Algebra Appl. 478 (2015), 218–227.
[14] A. M. Hasani and A. Ilkhanizadeh Manesh, Linear preservers of two-sided right matrix majorization on Rn, Adv. Oper. Theory, 3 (3) (2018), 1–8.
[15] A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly preserving majorizations of matrices, Elec. J. Linear Algeb. 15 (2006), 260–268.
[16] A. M. Hasani, Y. Sayyari and M. Sabzvari, G-tridiagonal majorization on Mn,m, Communications in Mathematics, 29 (3 (2021), 395 – 405.
[17] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of majorization and its applications, Springer, New York, 2011.
[18] Y. Sayyari, A. Mohammadhasani and M. Dehghanian, Linear maps preserving signed permutation and substochastic matrices, Indian J. Pure Appl. Math. 54 (2023), 219–223.

Files

History

  • Receive Date: 01 January 2023
  • Revise Date: 06 June 2023
  • Accept Date: 11 June 2023

Share

How to cite

Statistics