Strictly sub row Hadamard majorization

Document Type : Research Paper

Author

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

10.22103/jmmrc.2021.18576.1177

Abstract

‎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.

Keywords


[1] C. Davis, The norm of the Schur product operation, Numerische Mathematik, vol. 4, no 1. (1962) 343-344.
[2] B. Cyganek, Obeject detection and recognition in digital images (theory and practice), A John Wiley and Sons, 2013.
[3] P. H. George, Hadamard product and multivariate statistical analysis, Linear Algebra and its Applications vol. 6 (1973) 217-240 .
[4] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, 2012.
[5] S. M. Motlaghian, A. Armandnejad and F. J. Hall, Linear preservers of Hadamard majorization, Electronic Journal of Linear Algebra, vol 31. (2016) 593-609.