December 15, 2011 14:12 Linear and Multilinear Algebra rsmaj Linear and Multilinear Algebra Vol. 00, No. 00, Month 200x, 1–15 RESEARCH ARTICLE An extension of the polytope of doubly stochastic matrices Richard A. Brualdi a and Geir Dahl b∗ a Department of Mathematics, University of Wisconsin, Madison, WI 53706. b Center of Mathematics for Applications, Departments of Mathematics and Informatics, University of Oslo, P.O. Box 1053 Blindern, 0316 OSLO, Norway. (Received 00 Month 200x; in final form 00 Month 200x) We consider a class of matrices whose row and column sum vectors are majorized by given vectors b and c, and whose entries lie in the interval [0, 1]. This class generalizes the class of doubly stochastic matrices. We investigate the corresponding polytope Ω(b|c) of such ma- trices. Main results include a generalization of the Birkhoff - von Neumann theorem and a characterization of the faces, including edges, of Ω(b|c). Key words: Doubly stochastic matrices, majorization, polytope, faces. AMS subject classifications: 15B36, 15B51, 15A39. 1. Introduction Let Ω n denote the set of all doubly stochastic matrices of order n, i.e., nonnegative matrices where each row and column sum is 1. A classical theorem due to Birkhoff and von Neumann ([1], [9]) says that the extreme points of Ω n are the permuta- tion matrices. The purpose of this paper is to investigate a more general class of polytopes Ω(b|c) which contains Ω n as a special case. The underlying notion for defining Ω(b|c) is majorization. The i’th largest com- ponent in a vector x =(x 1 ,x 2 ,...,x n ) ∈ R n is denoted by x [i] . If x, y ∈ R n , we say that x is majorized by y, and write x  y, whenever ∑ k j =1 x [j ] ≤ ∑ k j =1 y [j ] for k =1, 2,...,n, with equality for k = n. Majorization plays an important role in e.g. combinatorics, statistics and matrix theory. The book [8] is a comprehensive study of majorization theory and its applications. The object we study, Ω(b|c), is the set of all matrices A =[a ij ] with 0 ≤ a ij ≤ 1 for each i, j and whose row sum vector and column sum vector satisfy a majorization constraint. The role of majorization in connection with classes of integral matrices or (0, 1)-matrices is discussed in detail in [4]. A central result is the Gale-Ryser theorem which characterizes the existence of a (0, 1)-matrix with given row and column sums in terms of a certain majorization for these given vectors. In [5] one studies doubly stochastic matrices whose rows and columns satisfy a majorization constraint, while [6] treats the class of integral matrices with given column sums and whose rows satisfy majorization constraints. The paper is organized as follows. Section 2 introduces the main notion, line- sum majorization and the class Ω(b|c). We relate this object to Ω n and prove a generalization of the Birkhoff - von Neumann theorem. In Section 3 the goal is to study the facial structure of the polytope Ω(b|c). A useful connection to so-called * Corresponding author. Email: geird@math.uio.no ISSN: 0308-1087 print/ISSN 1563-5139 online c  200x Taylor & Francis DOI: 10.1080/03081080xxxxxxxxx http://www.informaworld.com