# Article

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Keywords:
g-row stochastic matrix; gut-majorization; linear preserver; strong linear preserver; two-sided gut-majorization
Summary:
For $X,Y \in {\bf M}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X\prec _{\rm gut} Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation $\sim _{\rm gut}$ as follows. $X\sim _{\rm gut}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of $\prec _{\rm gut}$ on $\mathbb {R}^{n}$ and strong linear preservers of this relation on ${\bf M}_{n,m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of $\sim _{\rm gut}$ on $\mathbb {R}^{n}$ and ${\bf M}_{n,m}$.
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