A SHORT PROOF OF THE BIRKHOFF-VON NEUMANN THEOREM GLENN HURLBERT * Abstract. The Birkhoff-von Neumann Theorem has been proved many times in the literature with a number of different methods, some inductive, some constructive, some existential. We offer a new proof that is a little more direct than most, though nonconstructive. Key words. Doubly stochastic matrices, Convex combinations, Permutation matrices. AMS subject classifications. 15A51, 52A20. 1. Introduction. A vector is stochastic if it is nonnegative and its components sum to 1. A matrix is doubly stochastic (DS) if each of its rows and columns is stochastic. A permutation matrix is a square {0, 1}-matrix with exactly one 1 per row and per column. The identity matrix is an example of a permutation matrix; indeed, every permutation matrix is a rearrangement of the columns (or rows) of an identity matrix. A permutation matrix is one type of doubly stochastic matrix; in fact, every integral doubly stochastic matrix is a permutation matrix. It is elementary that every convex combination of permutation matrices is DS. The converse is a 1936 theorem of K˝ onig [7] (Chapter XIV, Section 3, in the context of generalizing the factorization of regular bipartite graphs), typically attributed instead to the 1946 and 1953 work of Birkhoff [2] and von Neumann [8], respectively. Theorem 1.1. Every DS matrix is a convex combination of permutation matri- ces. The traditional proof uses induction by removing an appropriate fraction of a permutation matrix P from the given DS matrix, and various methods have been found to find such a P , including von Neumann’s iterated scheme (similar to our method below) as well as linear optimization (see [3, 4, 6]) — essentially an application of the integrality theorem for networks. Edmonds’ proof (given on p. 331 of [3]) applies network theory more directly, instead of to the permutation matrix lemma. Another interesting proof is found in [9], reminscent of the Frobenius-K˝ onig theorem (see [1], p.62) characterising 0-permanent matrices. The proof in [5] uses induction directly to prove Theorem 1.1. Our proof is also direct, avoiding the permutation matrix lemma; however it is consequently nonconstructive. The motivation for this * Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287- 1804, USA (hurlbert@asu.edu). 1