v2007.09.17 - Convex Optimization

106 CHAPTER 2. CONVEX GEOMETRYwhich is isomorphic with the convex cone S rank A+ . Thus dimension of thesmallest face containing given matrix A isdim F ( S M + ∋A ) = rank(A)(rank(A) + 1)/2 (191)in isomorphic R M(M+1)/2 , and each and every face of S M + is isomorphic witha positive semidefinite cone having dimension the same as the face. Observe:not all dimensions are represented, and the only zero-dimensional face is theorigin. The positive semidefinite cone has no facets, for example.2.9.2.3.1 Table: Rank k versus dimension of S 3 + facesk dim F(S 3 + ∋ rank-k matrix)0 0boundary 1 12 3interior 3 6For the positive semidefinite cone S 2 + in isometrically isomorphic R 3depicted in Figure 31, for example, rank-2 matrices belong to the interior ofthe face having dimension 3 (the entire closed cone), while rank-1 matricesbelong to the relative interior of a face having dimension 1 (the boundaryconstitutes all the one-dimensional faces, in this dimension, which are raysemanating from the origin), and the only rank-0 matrix is the point at theorigin (the zero-dimensional face).Any simultaneously diagonalizable positive semidefinite rank-k matricesbelong to the same face (190). That observation leads to the followinghyperplane characterization of PSD cone faces: Any rank-k