v2010.10.26 - Convex Optimization

60 CHAPTER 2. CONVEX GEOMETRYbecause any matrix in subspace R M×Mhis orthogonal to any matrix in theantisymmetric antihollow subspace{ }1R M×M⊥h=2 (A −AT ) + δ 2 (A) | A∈ R M×M ⊆ R M×M (71)of the ambient space of real matrices; which reduces to the diagonal matricesin the ambient space of symmetric matricesS M⊥h = { δ 2 (A) | A∈ S M} = { δ(u) | u∈ R M} ⊆ S M (72)In anticipation of their utility with Euclidean distance matrices (EDMs)in5, for symmetric hollow matrices we introduce the linear bijectivevectorization dvec that is the natural analogue to symmetric matrixvectorization svec (56): for Y = [Y ij ]∈ S M h⎡dvec Y √ 2⎢⎣⎤Y 12Y 13Y 23Y 14Y 24Y 34 ⎥.⎦Y M−1,M∈ R M(M−1)/2 (73)Like svec , dvec is an isometric isomorphism on the symmetric hollowsubspace. For X ∈ S M h‖ dvec X − dvecY ‖ 2 = ‖X − Y ‖ F (74)The set of all symmetric hollow matrices S M h forms a proper subspacein R M×M , so for it there must be a standard orthonormal basis inisometrically isomorphic R M(M−1)/2{ }1 ( ){E ij ∈ S M h } = √ ei e T j + e j e T i , 1 ≤ i < j ≤ M (75)2where M(M −1)/2 standard basis matrices E ij are formed from the standardbasis vectors e i ∈ R M .The symmetric hollow majorization corollary A.1.2.0.2 characterizeseigenvalues of symmetric hollow matrices.