v2007.11.26 - Convex Optimization

5.4. EDM DEFINITION 3155.4.2 Gram-form EDM definitionPositive semidefinite matrix X T X in (734), formed from inner product oflist X , is known as a Gram matrix; [183,3.6]⎡⎤‖x 1 ‖ 2 x T⎡ ⎤1x 2 x T 1x 3 · · · x T 1x Nx T 1 [x 1 · · · x N ]xG = ∆ X T ⎢ ⎥T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x NX = ⎣ . ⎦ =x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N∈ S N +xNT ⎢⎥⎣ . ....... . ⎦xN Tx 1 xN Tx 2 xN Tx 3 · · · ‖x N ‖ 2⎡⎤⎛⎡⎤⎞1 cos ψ 12 cos ψ 13 · · · cos ψ 1N ⎛⎡⎤⎞‖x 1 ‖‖x ‖x 2 ‖cos ψ 12 1 cos ψ 23 · · · cos ψ 2N1 ‖= δ⎜⎢⎥⎟⎝⎣. ⎦⎠cos ψ 13 cos ψ 23 1... cos ψ ‖x 2 ‖3Nδ⎜⎢⎥⎟⎢⎥ ⎝⎣. ⎦⎠‖x N ‖⎣ . ....... . ⎦‖x N ‖cos ψ 1N cosψ 2N cos ψ 3N · · · 1∆= √ δ 2 (G) Ψ √ δ 2 (G) (743)where ψ ij (763) is angle between vectors x i and x j , and where δ 2 denotesa diagonal matrix in this case. Positive semidefiniteness of interpoint anglematrix Ψ implies positive semidefiniteness of Gram matrix G ; [46,8.3]G ≽ 0 ⇐ Ψ ≽ 0 (744)When δ 2 (G) is nonsingular, then G ≽ 0 ⇔ Ψ ≽ 0. (A.3.1.0.5)Distance-square d ij (730) is related to Gram matrix entries G T = G ∆ = [g ij ]d ij = g ii + g jj − 2g ij= 〈Φ ij , G〉(745)where Φ ij is defined in (732). Hence the linear EDM definition}D(G) = ∆ δ(G)1 T + 1δ(G) T − 2G ∈ EDM N⇐ G ≽ 0 (746)= [〈Φ ij , G〉 , i,j=1... N]The EDM cone may be described, (confer (823)(829))EDM N = { }D(G) | G ∈ S N +(747)