Linear transformations that preserve majorization, Schur concavity, and exchangeability
type: Item
instance of: scholarly article
title: Linear transformations that preserve majorization, Schur concavity, and exchangeability
zbMATH Open document ID: 0712.15002
DOI: 10.1016/0024-3795(90)90339-E
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: A vector \(w\in R^ n\) is said to majorize another vector \(x\in R^ n\) (notation: \(w>x)\) if \(\sum^{n}_{i=1}w_ i=\sum^{n}_{i-1}x_ i\) and \(\sum^{k}_{i=1}w_{[i]}\geq \sum^{k}_{i=1}x_{[i]}\) for each \(k=1,2,...,n-1\) where \(x_{[1]}\geq x_{[2]}\geq...\geq x_{[n]}\) denote the components of x in decreasing order. A random vector \(x\in R^ n\) is said to be exchangeable if R(\(\tau\))x has the same distribution as x for any permutation matrix R(\(\tau\)). A linear transformation A: \(R^ m\to R^ n\) is said to preserve majorization if \(Ay>Az\) whenever \(y>z.\) Let \(m\leq n\), A: \(R^ m\to R^ n\), B: \(R^ n\to R^ m\), A and B have rank m. Condition 1: for every permutation \(\pi\) there exists a permutation \(\tau\) such that \(R(\tau)A=AR(\pi).\) Condition 2: for every permutation \(\pi\) there exists a permutation \(\sigma\) such that \(BR(\sigma)=R(\pi)B.\) First the authors find the explicit forms of matrices that satisfy conditions 1 and 2. Then they prove several theorems that relate conditions 1 and 2 to majorization, exchangeability and Schur concavity, for example: the transformation A preserves majorization iff it satisfies condition 1 (Theorem 3); the transformation B preserves exchangeability iff it satisfies condition 2. Some of the results are generalized for the case when the group of permutations is replaced by its subgroup. Using the algebraic results the authors present some applications to multivariate density functions.
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 4171108
zbMATH Keywords: linear transformation
zbMATH Keywords: majorization
zbMATH Keywords: exchangeability
zbMATH Keywords: Schur concavity
zbMATH Keywords: multivariate density functions
author: A. M. Dean
author: Joseph Verducci
reviewed by: Boris Raykhshteyn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4171108