v2007.09.13 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 107makes projectors Q(:,k+1:M)Q(:,k+1:M) T indexed by k , in other words,each projector describing a normal svec ( Q(:,k+1:M)Q(:,k+1:M) T) to asupporting hyperplane ∂H + (containing the origin) exposing (2.11) a faceof the positive semidefinite cone containing only rank-k matrices.2.9.2.4 Extreme directions of positive semidefinite coneBecause the positive semidefinite cone is pointed (2.7.2.1.2), there is aone-to-one correspondence of one-dimensional faces with extreme directionsin any dimension M ; id est, because of the cone faces lemma (2.8.0.0.1)and the direct correspondence of exposed faces to faces of S M + , it followsthere is no one-dimensional face of the positive semidefinite cone that is nota ray emanating from the origin.Symmetric dyads constitute the set of all extreme directions: For M >0{yy T ∈ S M | y ∈ R M } ⊂ ∂S M + (193)this superset (confer (156)) of extreme directions for the positive semidefinitecone is, generally, a subset of the boundary. For two-dimensional matrices,(Figure 31){yy T ∈ S 2 | y ∈ R 2 } = ∂S 2 + (194)while for one-dimensional matrices, in exception, (2.7){yy T ∈ S | y≠0} = int S + (195)Each and every extreme direction yy T makes the same angle with theidentity matrix in isomorphic R M(M+1)/2 , dependent only on dimension;videlicet, 2.32(yy T , I) = arccos( )〈yy T , I〉 1= arccos √‖yy T ‖ F ‖I‖ F M∀y ∈ R M (196)2.32 Analogy with respect to the EDM cone is considered by Hayden & Wells et alii[133, p.162] where it is found: angle is not constant. The extreme directions of the EDMcone can be found in6.5.3.1 while the cone axis is −E =11 T − I (898).