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Linear Operators Preserving Majorization of Matrix Tuples
Vestnik St. Petersburg University, Mathematics Pub Date : 2020-06-02 , DOI: 10.1134/s1063454120020077
A. E. Guterman , P. M. Shteyner

Abstract

In this paper, we consider weak, directional and strong matrix majorizations. Namely, for square matrices A and B of the same size we say that A is weakly majorized by B if there is a row stochastic matrix X such that A = XB. Further, A is strongly majorized by B if there is a doubly stochastic matrix X such that A = XB. Finally, A is directionally majorized by B if Ax is majorized by Bx for any vector x where the usual vector majorization is used. We introduce the notion of majorization of matrix tuples which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one tuple of matrices is majorized by another tuple of the same size if every matrix of the “smaller” tuple is majorized by a matrix in the same position in the “bigger” tuple. We say that a linear operator preserves majorization if it maps ordered pairs to ordered pairs and the image of the smaller element does not exceed the image of the bigger one. This paper contains a full characterization of linear operators that preserve weak, strong or directional majorization of tuples of matrices and linear operators that map tuples that are ordered with respect to strong majorization to tuples that are ordered with respect to directional majorization. We have shown that every such operator preserves respective majorization of each component. For all types of majorization we provide counterexamples that demonstrate that the inverse statement does not hold, that is if majorization of each component is preserved, majorization of tuples may not.



中文翻译:

保留矩阵元组主化的线性算子

摘要

在本文中,我们考虑了弱,方向性和强矩阵主化。即,对于方阵的大小相同我们说弱通过majorized如果有一个行随机矩阵X,使得= XB。此外,强烈地受到majorized如果有一个双随机矩阵X,使得= XB。最后,如果对于任何向量x,如果AxBx主化,则AB定向主化。使用通常的矢量主化的地方。我们引入矩阵元组的主化概念,该概念定义为矩阵主化的自然概括:对于所选类型的主化,我们说,如果每个“较小”的矩阵都由一个相同大小的元组主化。元组由位于“较大”元组中相同位置的矩阵主化。我们说如果线性算子将有序对映射到有序对并且较小元素的图像不超过较大元素的图像,则保留主化。本文全面描述了保留弱,矩阵和线性运算符的元组的强或方向主化,这些映射将关于强主化的有序元组映射到关于方向主化的有序元组。我们已经表明,每个这样的运算符都会保留每个组成部分的相应主化。对于所有类型的主化,我们提供了反例来证明逆语句不成立,也就是说,如果保留每个组件的主化,则元组的主化可能不会。

更新日期:2020-06-02
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