v2009.01.01 - Convex Optimization

418 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
D ∈ EDM N ⇒
⎧
λ(−D) i ≥ 0, i=1... N −1
⎪⎨ ( N
)
∑
λ(−D) i = 0
i=1
⎪⎩
D ∈ S N h ∩ R N×N
+
(1009)
where the λ(−D) i are nonincreasingly ordered eigenvalues of −D whose
sum must be 0 only because trD = 0 [287,5.1]. The eigenvalue summation
condition, therefore, can be considered redundant. Even so, all these
conditions are insufficient to determine whether some given H ∈ S N h is an
EDM, as shown by counter-example. 5.47
5.11.1.0.1 Exercise. Spectral inequality.
Prove whether it holds: for D=[d ij ]∈ EDM N
λ(−D) 1 ≥ d ij ≥ λ(−D) N−1 ∀i ≠ j (1010)
Terminology: a spectral cone is a convex cone containing all eigenspectra
[197, p.365] [285, p.26] corresponding to some set of matrices. Any
positive semidefinite matrix, for example, possesses a vector of nonnegative
eigenvalues corresponding to an eigenspectrum contained in a spectral cone
that is a nonnegative orthant.
5.11.2 Spectral cones
[ 0 1
T
Denoting the eigenvalues of Cayley-Menger matrix
1 −D
([ 0 1
T
λ
1 −D
5.47 When N = 3, for example, the symmetric hollow matrix
H =
⎡
⎣ 0 1 1
1 0 5
1 5 0
⎤
]
∈ S N+1 by
])
∈ R N+1 (1011)
⎦ ∈ S N h ∩ R N×N
+
is not an EDM, although λ(−H) = [5 0.3723 −5.3723] T conforms to (1009).