A Birkhoff Theorem for Doubly Stochastic Matrices with Vector Entries

Abstract

The concept of a doubly stochastic matrix whose entries come from a convex subset of the unit square is defined. It is proved that the only convex subsets of the unit square which contain (0,0) and (1,1) and allow an extension of Birkhoff's characterization of the extreme points of the set of doubly stochastic matrices are parallelograms. A sufficient condition is given for a matrix to be extreme when the convex subset is not a parallelogram. o. IntroductionAn n X n matrix whose entries are taken from the closed unit interval and whose rows and columns sum to one was called doubly stochastic by Feller [9]. The importance of these matrices to applications of matrix theory goes back at least to Schur's work in 1923 [15], and their study has produced an abundance of theoretical papers which in turn have generated fertile areas of research. Two such important works are Hardy, Littlewood, and Polya's paper [10] The set of n X n doubly stochastic matrices is the convex hull of the permutation matrices (those with I in each row and column).Then in problem Ill, he suggests this theorem be generalized to the infinite case. The work of J. B. Isbell [11] and D. G. Kendall [12] concerns that countable case; that of J. Lindenstrauss [13], J. V. Ryff [14], and Brown and Shiflett [3] concerns the continuous case. Investigations continue on questions of extreme points of convex subsets of this set of matrices. Of particular interest are the papers by R. Brualdi [4][5][6][7][8]. It is proposed here that a new and apparently overlooked type of doubly stochastic matrix be considered which may be of