v2007.11.26 - Convex Optimization

2.3. HULLS 53Figure 14: Convex hull of a random list of points in R 3 . Some pointsfrom that generating list reside in the interior of this convex polyhedron.[284, Convex Polyhedron] (Avis-Fukuda-Mizukoshi)The set of all symmetric hollow matrices S M h forms a proper subspace inR M×M , so for it there must be a standard orthonormal basis in isometricallyisomorphic R M(M−1)/2{E ij ∈ S M h } ={ }1 ( )√ ei e T j + e j e T i , 1 ≤ i < j ≤ M 2(65)where M(M −1)/2 standard basis matrices E ij are formed from the standardbasis vectors e i ∈ R M .The symmetric hollow majorization corollary on page 498 characterizeseigenvalues of symmetric hollow matrices.2.3 Hulls2.3.1 Affine hull, affine dimensionAffine dimension of any set in R n is the dimension of the smallest affineset (empty set, point, line, plane, hyperplane (2.4.2), subspace, R n ) thatcontains it. For nonempty sets, affine dimension is the same as dimension ofthe subspace parallel to that affine set. [232,1] [148,A.2.1]Ascribe the points in a list {x l ∈ R n , l=1... N} to the columns ofmatrix X :X = [x 1 · · · x N ] ∈ R n×N (66)