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

Document Type: Original Article


Rafsanjan University of Vali Asr


A nonnegative real matrix $R\in \textbf{M}_{n,m}$ 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 \textbf{1} (2014), no. 1, 43--50].