On extreme infinite doubly stochastic matrices
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.
scientific article; zbMATH DE number 4014911