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.
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Received July 8, 1999, and in revised form January 20, 2000, and May 9, 2000. Online publication September 22, 2000.
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Barvinok, A. A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints . Discrete Comput Geom 25, 23–31 (2001). https://doi.org/10.1007/s004540010074
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DOI: https://doi.org/10.1007/s004540010074