A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints

Abstract

Let K n be the cone of positive semidefinite n X n matrices and let Å be an affine subspace of the space of symmetric matrices such that the intersection K n ∩Å is nonempty and bounded. Suppose that n ≥ 3 and that \codim Å = r+2 \choose 2 for some 1 ≤ r ≤ n-2 . Then there is a matrix X ∈ K n ∩Å such that rank X ≤ r . We give a short geometric proof of this result, use it to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and describe its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.