v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 355
5.4.2 Gram-form EDM definition
Positive semidefinite matrix X T X in (798), formed from inner product of
list X , is known as a Gram matrix; [215,3.6]
⎡
⎤
‖x 1 ‖ 2 x T
⎡ ⎤
1x 2 x T 1x 3 · · · x T 1x N
x T 1 [ x 1 · · · x N ]
x
G = ∆ X T ⎢ ⎥
T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x N
X = ⎣ . ⎦ =
x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N
∈ S N +
xN
T ⎢
⎥
⎣ . .
...
... . ⎦
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 2N
1 ‖
= δ⎜⎢
⎥⎟
⎝⎣
. ⎦⎠
cos ψ 13 cos ψ 23 1
... cos ψ ‖x 2 ‖
3N
δ⎜⎢
⎥⎟
⎢
⎥ ⎝⎣
. ⎦⎠
‖x N ‖
⎣ . .
...
... . ⎦
‖x N ‖
cos ψ 1N cosψ 2N cos ψ 3N · · · 1
∆
= √ δ 2 (G) Ψ √ δ 2 (G) (807)
where ψ ij (827) is angle between vectors x i and x j , and where δ 2 denotes
a diagonal matrix in this case. Positive semidefiniteness of interpoint angle
matrix Ψ implies positive semidefiniteness of Gram matrix G ; [53,8.3]
G ≽ 0 ⇐ Ψ ≽ 0 (808)
When δ 2 (G) is nonsingular, then G ≽ 0 ⇔ Ψ ≽ 0. (A.3.1.0.5)
Distance-square d ij (794) is related to Gram matrix entries G T = G ∆ = [g ij ]
d ij = g ii + g jj − 2g ij
= 〈Φ ij , G〉
(809)
where Φ ij is defined in (796). Hence the linear EDM definition
}
D(G) = ∆ δ(G)1 T + 1δ(G) T − 2G ∈ EDM N
⇐ G ≽ 0 (810)
= [〈Φ ij , G〉 , i,j=1... N]
The EDM cone may be described, (confer (894)(900))
EDM N = { }
D(G) | G ∈ S N +
(811)