v2007.09.13 - Convex Optimization

340 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXFor all practical purposes, (846)max{0, rank(D) − 2} ≤ r ≤ min{n, N −1} (855)5.8 Euclidean metric versus matrix criteria5.8.1 Nonnegativity property 1When D=[d ij ] is an EDM (705), then it is apparent from (840)2VNX T T XV N = −VNDV T N ≽ 0 (856)because for any matrix A , A T A≽0 . 5.32 We claim nonnegativity of the d ijis enforced primarily by the matrix inequality (856); id est,−V T N DV N ≽ 0D ∈ S N h}⇒ d ij ≥ 0, i ≠ j (857)(The matrix inequality to enforce strict positivity differs by a stroke of thepen. (860))We now support our claim: If any matrix A∈ R m×m is positivesemidefinite, then its main diagonal δ(A)∈ R m must have all nonnegativeentries. [109,4.2] Given D ∈ S N h−VN TDV N =⎡⎤1d 12 2 (d 112+d 13 −d 23 )2 (d 11,i+1+d 1,j+1 −d i+1,j+1 ) · · ·2 (d 12+d 1N −d 2N )12 (d 112+d 13 −d 23 ) d 13 2 (d 11,i+1+d 1,j+1 −d i+1,j+1 ) · · ·2 (d 13+d 1N −d 3N )1⎢2 (d 11,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 .. 11,i+1 2 (d 14+d 1N −d 4N )⎥⎣..... . .. . ⎦12 (d 112+d 1N −d 2N )2 (d 113+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 (858)5.32 For A∈R m×n , A T A ≽ 0 ⇔ y T A T Ay = ‖Ay‖ 2 ≥ 0 for all ‖y‖ = 1. When A isfull-rank skinny-or-square, A T A ≻ 0.