v2007.11.26 - Convex Optimization

6.6. CORRESPONDENCE TO PSD CONE S N−1+ 4256.6 Correspondence to PSD cone S N−1+Hayden & Wells et alii [134,2] assert one-to-one correspondence of EDMswith positive semidefinite matrices in the symmetric subspace. Becauserank(V DV )≤N−1 (5.7.1.1), that positive semidefinite cone correspondingto the EDM cone can only be S N−1+ . [6,18.2.1] To clearly demonstrate thiscorrespondence, we invoke inner-product form EDM definition[ ]D(Φ) =∆ 01δ(Φ)T + 1 [ 0 δ(Φ) ] [ ] 0 0 T T− 2∈ EDM N0 Φ⇔(851)Φ ≽ 0Then the EDM cone may be expressedEDM N = { D(Φ) | Φ ∈ S N−1+}(1049)Hayden & Wells’ assertion can therefore be equivalently stated in terms ofan inner-product form EDM operatorD(S N−1+ ) = EDM N (853)V N (EDM N ) = S N−1+ (854)identity (854) holding because R(V N )= N(1 T ) (741), linear functions D(Φ)and V N (D)= −VN TDV N (5.6.2.1) being mutually inverse.In terms of affine dimension r , Hayden & Wells claim particularcorrespondence between PSD and EDM cones:r = N −1: Symmetric hollow matrices −D positive definite on N(1 T ) correspondto points relatively interior to the EDM cone.r < N −1: Symmetric hollow matrices −D positive semidefinite on N(1 T ) , where−VN TDV N has at least one 0 eigenvalue, correspond to points on therelative boundary of the EDM cone.r = 1: Symmetric hollow nonnegative matrices rank-one on N(1 T ) correspondto extreme directions (1041) of the EDM cone; id est, for some nonzerovector u (A.3.1.0.7)rankV T N DV N =1D ∈ S N h ∩ R N×N+}⇔{D ∈ EDM N−VTD is an extreme direction ⇔ N DV N ≡ uu TD ∈ S N h(1050)