The doubly stochastic matrices of a vector majorization
type: Item
instance of: scholarly article
title: The doubly stochastic matrices of a vector majorization
zbMATH Open document ID: 0558.15008
DOI: 10.1016/0024-3795(84)90027-2
author: Richard A. Brualdi
published in: Linear Algebra and its Applications
publication date: 1984-01-01T00:00:00Z
review text: Let x and y be real n-dimensional column vectors; y is said to be majorized by x if the following holds: if \(x_{[1]},...,x_{[n]}\) are the components of x arranged in nonincreasing order and \(y_{[1]},...,y_{[n]}\) are likewise for y, then \(\sum^{k}_{i=1}y_{[i]}\leq \sum^{k}_{i=1}x_{[i]},\) \(k=1,...,n\), with equality for \(k=n\). The polytope of doubly stochastic matrices D for which \(y=Dx\) is investigated. It is determined when there exists a positive D and when there exists a fully indecomposable D. The dimension of the polytope is computed and as a consequence it is determined just when D is unique.
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 3889725
zbMATH Keywords: vector majorization
zbMATH Keywords: polytope of doubly stochastic matrices
zbMATH Keywords: fully indecomposable
zbMATH Keywords: dimension
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
full work available at URL: 0024-3795(84)90027-2
OpenAlex ID: W2068986444
cites work: Convex polyhedra of doubly stochastic matrices. I: Applications of the permanent function
cites work: Q5812325
cites work: A problem of Mirsky concerning nonsingular doubly stochastic matrices
cites work: Inequalities: theory of majorization and its applications
cites work: Q3851094
description: scientific article; zbMATH DE number 3889725