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For it is said that is gut-majorized by , and we write , if there exists an -by- upper triangular g-row stochastic matrix such that . Define the relation as follows. if is gut-majorized by and is gut-majorized by . The (strong) linear preservers of on and strong linear preservers of this relation on have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of on and .
For , it is said that is majorized by (and it is denoted by ) if there exists a tridiagonal g-doubly stochastic matrix such that . In this paper, the linear preservers and strong linear preservers of are characterized on .
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