Geodesic
Row Hadamard majorization on ${\bf M}_{m,n}$
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 743-754
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An $m \times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let ${\bf M}_{m,n}$ be the set of all $m \times n$ real matrices. For $A,B\in \nobreak {\bf M}_{m,n}$, we say that $A$ is row Hadamard majorized by $B$ (denoted by $A\prec _{RH}B)$ if there exists an $m \times n$ row stochastic matrix $R$ such that $A=R\circ B$, where $X \circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in {\bf M}_{m,n}$. In this paper, we consider the concept of row Hadamard majorization as a relation on ${\bf M}_{m,n}$ and characterize the structure of all linear operators $T\colon {\bf M}_{m,n} \rightarrow {\bf M}_{m,n}$ preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on ${\bf M}_{n}$.
An $m \times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let ${\bf M}_{m,n}$ be the set of all $m \times n$ real matrices. For $A,B\in \nobreak {\bf M}_{m,n}$, we say that $A$ is row Hadamard majorized by $B$ (denoted by $A\prec _{RH}B)$ if there exists an $m \times n$ row stochastic matrix $R$ such that $A=R\circ B$, where $X \circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in {\bf M}_{m,n}$. In this paper, we consider the concept of row Hadamard majorization as a relation on ${\bf M}_{m,n}$ and characterize the structure of all linear operators $T\colon {\bf M}_{m,n} \rightarrow {\bf M}_{m,n}$ preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on ${\bf M}_{n}$.
DOI : 10.21136/CMJ.2020.0081-20
Classification : 15A04, 15A21
Keywords: linear preserver; row Hadamard majorization; row stochastic matrix 
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Askarizadeh, Abbas; Armandnejad, Ali. Row Hadamard majorization on ${\bf M}_{m,n}$. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 743-754. doi: 10.21136/CMJ.2020.0081-20

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