### Linear preservers of two-sided right matrix majorization on $\mathbb{R}_{n}$

#### Abstract

A nonnegative real matrix $R \in \mathrm {M}_{n,m}$ with the property that all its row sums are one is said to be row stochastic. For $x, y \in \mathbb{R}_{n}$, we say $x$ is right matrix majorized by $y$ (denoted by $x \prec_{r} y$) if there exists an $n$-by-$n$ row stochastic matrix $R$ such that $x = yR$. The relation $\sim_{r}$ on $\mathbb{R}_{n}$ is defined as follows. $x \sim_{r}y$ if and only if $x \prec_{r} y \prec_{r} x$. In the present paper, we characterize the linear preservers of $\sim_{r}$ on $\mathbb{R}_{n}$, and answer the question raised by F. Khalooei [Wavelet Linear Algebra 1 (2014), no. 1, 43-50].

#### Article information

Source
Adv. Oper. Theory , Number (2018), 9 pages.

Dates
Accepted: 3 December 2017
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.aot/1513876632

Digital Object Identifier
doi:10.15352/aot.1709-1225

Subjects
Primary: 15A04: Linear transformations, semilinear transformations
Secondary: 15A51

#### Citation

Mohammadhasani, Ahmad; Ilkhanizadeh Manesh, Asma. Linear preservers of two-sided right matrix majorization on $\mathbb{R}_{n}$. Adv. Oper. Theory, advance publication, 21 December 2017. doi:10.15352/aot.1709-1225. https://projecteuclid.org/euclid.aot/1513876632

#### References

• T. Ando, Majorization, doubly stochastic matrices, and comparision of eigenvalues, Linear Algebra Appl. 118 (1989), 163–248.
• H. Chiang and C.-K. Li, Generalized doubly stochastic matrices and linear preservers, Linear Multilinear Algebra 53 (2005), 1–11.
• A. M. Hasani and M. Radjabalipour, The structure of linear operators strongly preserving majorizations of matrices, Electron. J. Linear Algebra 15 (2006), 260–268.
• A. M. Hasani and M. Radjabalipour, On linear preservers of (right) matrix majorization, Linear Algebra Appl 423 (2007), 255–261.
• A. Ilkhanizadeh Manesh, Right gut-Majorization on $\textbf{M}_{n,m}$, Electron. J. Linear Algebra 31 (2016), no. 1, 13–26.
• F. Khalooei, Linear preservers of two-sided matrix majorization, Wavelet Linear Algebra 1 (2014), no. 1, 43–50.
• A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: theory of majorization and its applications, Second edition. Springer Series in Statistics, Springer, New York, 2011.