v2009.01.01 - Convex Optimization

398 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
For all practical purposes, (946)
max{0, rank(D) − 2} ≤ r ≤ min{n, N −1} (955)
5.8 Euclidean metric versus matrix criteria
5.8.1 Nonnegativity property 1
When D=[d ij ] is an EDM (798), then it is apparent from (940)
2VNX T T XV N = −VNDV T N ≽ 0 (956)
because for any matrix A , A T A≽0 . 5.33 We claim nonnegativity of the d ij
is enforced primarily by the matrix inequality (956); id est,
−V T N DV N ≽ 0
D ∈ S N h
}
⇒ d ij ≥ 0, i ≠ j (957)
(The matrix inequality to enforce strict positivity differs by a stroke of the
pen. (960))
We now support our claim: If any matrix A∈ R m×m is positive
semidefinite, then its main diagonal δ(A)∈ R m must have all nonnegative
entries. [134,4.2] Given D ∈ S N h
−VN TDV N =
⎡
⎤
1
d 12 2 (d 1
12+d 13 −d 23 )
2 (d 1
1,i+1+d 1,j+1 −d i+1,j+1 ) · · ·
2 (d 12+d 1N −d 2N )
1
2 (d 1
12+d 13 −d 23 ) d 13 2 (d 1
1,i+1+d 1,j+1 −d i+1,j+1 ) · · ·
2 (d 13+d 1N −d 3N )
1
⎢
2 (d 1
1,j+1+d 1,i+1 −d j+1,i+1 )
2 (d .
1,j+1+d 1,i+1 −d j+1,i+1 ) d .. 1
1,i+1 2 (d 14+d 1N −d 4N )
⎥
⎣
.
.
.
.. . .. . ⎦
1
2 (d 1
12+d 1N −d 2N )
2 (d 1
13+d 1N −d 3N )
2 (d 14+d 1N −d 4N ) · · · d 1N
= 1 2 (1D 1,2:N + D 2:N,1 1 T − D 2:N,2:N ) ∈ S N−1 (958)
5.33 For A∈R m×n , A T A ≽ 0 ⇔ y T A T Ay = ‖Ay‖ 2 ≥ 0 for all ‖y‖ = 1. When A is
full-rank skinny-or-square, A T A ≻ 0.