v2010.10.26 - Convex Optimization

470 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+(1104)where the λ(−D) i are nonincreasingly ordered eigenvalues of −D whosesum must be 0 only because trD = 0 [331,5.1]. 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 counterexample. 5.515.11.1.0.1 Exercise. Spectral inequality.Prove whether it holds: for D=[d ij ]∈ EDM Nλ(−D) 1 ≥ d ij ≥ λ(−D) N−1 ∀i ≠ j (1105)Terminology: a spectral cone is a convex cone containing all eigenspectra[227, p.365] [328, p.26] corresponding to some set of matrices. Anypositive semidefinite matrix, for example, possesses a vector of nonnegativeeigenvalues corresponding to an eigenspectrum contained in a spectral conethat is a nonnegative orthant.5.11.2 Spectral cones[ 0 1TDenoting the eigenvalues of Cayley-Menger matrix1 −D([ 0 1Tλ1 −D5.51 When N = 3, for example, the symmetric hollow matrixH =⎡⎣ 0 1 11 0 51 5 0⎤]∈ S N+1 by])∈ R N+1 (1106)⎦ ∈ S N h ∩ R N×N+is not an EDM, although λ(−H) = [5 0.3723 −5.3723] T conforms to (1104).