v2007.09.13 - Convex Optimization

360 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXD ∈ EDM N ⇒⎧λ(−D) i ≥ 0, i=1... N −1⎪⎨ ( N)∑λ(−D) i = 0i=1⎪⎩D ∈ S N h ∩ R N×N+(907)where the λ(−D) i are nonincreasingly ordered eigenvalues of −D whosesum must be 0 only because trD = 0 [247,5.1]. The eigenvalue summationcondition, therefore, can be considered redundant. Even so, all theseconditions are insufficient to determine whether some given H ∈ S N h is anEDM, as shown by counter-example. 5.465.11.1.0.1 Exercise. Spectral inequality.Prove whether it holds: for D=[d ij ]∈ EDM Nλ(−D) 1 ≥ d ij ≥ λ(−D) N−1 ∀i ≠ j (908)Terminology: a spectral cone is a convex cone containing all eigenspectracorresponding to some set of matrices. Any positive semidefinite matrix, forexample, possesses a vector of nonnegative eigenvalues corresponding to aneigenspectrum contained in a spectral cone that is a nonnegative orthant.5.11.2 Spectral cones[ 0 1TDenoting the eigenvalues of Cayley-Menger matrix1 −D([ 0 1Tλ1 −D]∈ S N+1 by])∈ R N+1 (909)we have the Cayley-Menger form (5.7.3.0.1) of necessary and sufficientconditions for D ∈ EDM N from the literature: [132,3] 5.47 [53,3] [77,6.2]5.46 When N = 3, for example, the symmetric hollow matrix⎡ ⎤0 1 1H = ⎣ 1 0 5 ⎦ ∈ S N h ∩ R+N×N1 5 0is not an EDM, although λ(−H) = [5 0.3723 −5.3723] T conforms to (907).5.47 Recall: for D ∈ S N h , −V T N DV N ≽ 0 subsumes nonnegativity property 1 (5.8.1).