A characterization of strong preservers of matrix majorization
type: Item
instance of: scholarly article
title: A characterization of strong preservers of matrix majorization
zbMATH Open document ID: 1032.15002
DOI: 10.1016/S0024-3795(02)00657-2
author: Sang-Gu Lee
author: Youho Lee
author: Leroy B. Beasley
published in: Linear Algebra and its Applications
publication date: 2003-05-27T00:00:00Z
review text: The definition of matrix majorization was introduced by \textit{G. Dahl} [Linear Algebra Appl. 288, 53-73 (1999; Zbl 0930.15023)] given several characterizations. Let \(A\) and \(B\) be \(m\times n\) real matrices. The matrix \(A\) is said to be matrix majorized by \(B\), \(A\prec B\), if there is an \(n\times n\) row stochastic matrix \(X\) (a nonnegative real matrix in which each of its row sums are equal to one) such that \(A=BX\). This definition generalizes the concept of multivariate majorization and is the matrix version of vector majorization that plays an important role in statistics and economics. In this paper, the authors characterize linear operators \(T\) that strongly preserve matrix majorization (\(T(A)\prec T(B)\) iff \(A\prec B\)), extending the results of \textit{L. B. Beasley} and \textit{S.-G. Lee} [Linear Algebra Appl. 304, 141-159 (2000; Zbl 0954.15016)] over the characterization of linear operators that strongly preserve multivariate majorization. The main theorem is: If \(T:M_{n}(\mathbb{R})\rightarrow M_{n}(\mathbb{R})\) strongly preserves matrix majorization, then there exists a permutation matrix \(Q\) and an invertible matrix \(M\) such that \(T(A)=MAQ\) for every \(A\) belonging to the span of the set of all row stochastic matrices.
reviewed by: M. Ladra
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1917823
zbMATH Keywords: matrix majorization
zbMATH Keywords: row stochastic matrix
zbMATH Keywords: linear preserver
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1917823