On extreme infinite doubly stochastic matrices
type: Item
instance of: scholarly article
title: On extreme infinite doubly stochastic matrices
zbMATH Open document ID: 0625.15013
author: Ryszard Grząślewicz
published in: Illinois Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \(\bar r=(r_ 1\), \(r_ 2\),...) and \(\bar s=(s_ 1\), \(s_ 2\),...) be given sequences of nonnegative real numbers. Matrices \(P=(p_{ij})\) with all \(p_{ij}\geq 0\), \(\sum^{\infty}_{k=1}p_{ik}\leq r_ i\) and \(\sum^{\infty}_{k=1}p_{kj}\leq s_ j\) are said to be doubly substochastic matrices with respect to \((\bar r,\bar s)\). The set of such matrices is denoted by \(D(\leq \bar r,\leq \bar s)\). If \(\sum^{\infty}_{i=1}r_ i=\sum^{\infty}_{j=1}s_ j\leq \infty\) and if all \(\sum^{\infty}_{k=1}p_{ik}=r_ i\) and \(\sum^{\infty}_{k=1}p_{kj}=s_ j\) these matrices are said to be doubly stochastic with respect to \((\bar r,\bar s)\). The set of such matrices is denotes by \(D(\bar r,\bar s)\). The sets \(D(\leq \bar r,\leq \bar s)\) and \(D(\bar r,\bar s)\) are each convex. It is shown that if \(\sum^{\infty}_{i=1}r_ i=\sum^{\infty}_{j=1}s_ j<\infty\), then \(P\in D(\bar r,\bar s)\) is extreme if and only if the connected components of the graph of P are trees. The extreme points of \(D(\bar r,\bar s)\) are precisely the exposed points in \(D(\bar r,\bar s)\). A matrix \(P\in D(\leq \bar r,\leq \bar s)\) is extreme if and only if the connected components of the graph of P are extreme trees. Some results are given concerning the dimension of \(D(\leq \bar r,\leq \bar s)\) when each of the sequences \(\bar r\) and \(\bar s\) is finite.
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 4014911
zbMATH Keywords: doubly substochastic matrices
zbMATH Keywords: connected components of the graph
zbMATH Keywords: trees
zbMATH Keywords: extreme points
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4014911