# 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

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### History

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