A note on symmetric doubly-stochastic matrices

Abstract

Each extreme point in the convex set Δ∗n of all n×n symmetric doubly-stochastic matrices is shown to have the form 12(P + Pt), for some n×n permutation matrix P. The convex hull Σn of the integral points in Δ∗n (i.e., the symmetric permutation matrices) is shown to consist of those matrices, X = (xij) in Δ∗n satisfying Σi∈SΣj∈S−{i}xij ⩽ 2k, for each subset S of {1, 2, …, n} having cardinality 2k + 1, for some k > 0.