Let \({\mathbf {M}}_{n,m}\) be the set of all n-by-m real matrices and let the relation \(\prec \) on \({\mathbf {M}}_{n,m}\) be the multivariate majorization. A linear mapping \(\Phi \!:{\mathbf {M}}_{n,m}\longrightarrow {\mathbf {M}}_{n,m}\) is said to be a linear preserver of multivariate majorization if \(\Phi (X)\prec \Phi (Y)\) whenever \(X\prec Y\). Also, \(\Phi \) is said to be a local linear preserver of multivariate majorization if for each \(X\in {\mathbf {M}}_{n,m}\) there exists a linear mapping \(\Phi _X\) on \({\mathbf {M}}_{n,m}\) that preserves, multivariate majorization and \(\Phi (X)=\Phi _X(X)\). In this paper we introduce two types of strong local linear preservers of multivariate majorization on \({\mathbf {M}}_{n,m}\). Then we find a relation between these types of strong local linear preservers and finally we characterize the structure of these linear maps.

中文翻译：

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多元主化的强局部线性保存器

令\（{\ mathbf {M}} _ {n，m} \）为所有n × m实矩阵的集合，并令\（{\ mathbf {M}}上的关系\（\ prec \）_ {n，m} \）是多元主化。线性映射\（\ Phi \ !: {\ mathbf {M}} _ {n，m} \ longrightarrow {\ mathbf {M}} _ {n，m} \）被认为是多元主化的线性保留如果\（\ Phi（X）\ prec \ Phi（Y）\）每当\（X \ prec Y \）。另外，如果每个{{\ mathbf {M}} _ {n，m} \）中存在一个线性映射\ {\，则\（\ Phi \）被认为是多元主化的局部线性保留者。披_X \）上\（{\ mathbf {M}} _ {n，m} \）保留，多元主化和\（\ Phi（X）= \ Phi _X（X）\）。在本文中，我们介绍了\（{\ mathbf {M}} _ {n，m} \）上的两种类型的多元主化的强局部线性保存器。然后，我们找到了这些类型的强局部线性保存器之间的关系，最后描述了这些线性图的结构。