v2010.10.26 - Convex Optimization

62 CHAPTER 2. CONVEX GEOMETRYfor which we call list X a set of generators. Hull A is parallel to subspacewhereR{x l − x 1 , l=2... N} = R(X − x 1 1 T ) ⊆ R n (79)R(A) = {Ax | ∀x} (142)Given some arbitrary set C and any x∈ Cwhere aff(C−x) is a subspace.aff C = x + aff(C − x) (80)2.3.1.0.1 Definition. Affine subset.We analogize affine subset to subspace, 2.14 defining it to be any nonemptyaffine set (2.1.4).△The affine hull of a point x is that point itself;aff ∅ ∅ (81)aff{x} = {x} (82)Affine hull of two distinct points is the unique line through them. (Figure 21)The affine hull of three noncollinear points in any dimension is that uniqueplane containing the points, and so on. The subspace of symmetric matricesS m is the affine hull of the cone of positive semidefinite matrices; (2.9)aff S m + = S m (83)2.3.1.0.2 Example. Affine hull of rank-1 correlation matrices. [218]The set of all m ×m rank-1 correlation matrices is defined by all the binaryvectors y in R m (confer5.9.1.0.1){yy T ∈ S m + | δ(yy T )=1} (84)Affine hull of the rank-1 correlation matrices is equal to the set of normalizedsymmetric matrices; id est,aff{yy T ∈ S m + | δ(yy T )=1} = {A∈ S m | δ(A)=1} (85)2.14 The popular term affine subspace is an oxymoron.