Strictly sub row Hadamard majorization

Document Type : Research Paper


Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran



‎Let $\textbf{M}_{m,n}$ be the set of all $m$-by-$n$ real matrices‎. ‎A matrix $R$ in $\textbf{M}_{m,n}$ with nonnegative entries is called strictly sub row stochastic if the sum of entries on every row of $R$ is less than 1‎. ‎For $A,B\in\textbf{M}_{m,n}$‎, ‎we say that $A$ is strictly sub row Hadamard majorized by $B$ (denoted by $A\prec_{SH}B)$ if there exists an $m$-by-$n$ strictly sub row stochastic matrix $R$ such that $A=R\circ B$ where $X \circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in\textbf{M}_{m,n}$‎. ‎In this paper‎, ‎we introduce the concept of strictly sub row Hadamard majorization as a relation on $\textbf{M}_{m,n}$‎. ‎Also‎, ‎we find the structure of all linear operators $T:\textbf{M}_{m,n} \rightarrow \textbf{M}_{m,n}$ which are preservers (resp‎. ‎strong preservers) of strictly sub row Hadamard majorization‎.


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