v2007.09.17 - Convex Optimization

52 CHAPTER 2. CONVEX GEOMETRYAny matrix A∈ R M×M can be written as the sum of its symmetric hollowand antisymmetric antihollow parts: respectively,( ) ( 1 1A =2 (A +AT ) − δ 2 (A) +2 (A −AT ) + δ (A))2 (59)The symmetric hollow part is orthogonal in R M2 to the antisymmetricantihollow part; videlicet,))1 1tr((2 (A +AT ) − δ (A))( 2 2 (A −AT ) + δ 2 (A) = 0 (60)In the ambient space of real matrices, the antisymmetric antihollow subspaceis described{ }∆ 1=2 (A −AT ) + δ 2 (A) | A∈ R M×M ⊆ R M×M (61)S M⊥hbecause any matrix in S M h is orthogonal to any matrix in S M⊥h . Yet inthe ambient space of symmetric matrices S M , the antihollow subspace isnontrivial;S M⊥h∆= { δ 2 (A) | A∈ S M} = { δ(u) | u∈ R M} ⊆ S M (62)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 (47): for Y = [Y ij ]∈ S M h⎡dvecY = ∆ √ 2⎢⎣⎤Y 12Y 13Y 23Y 14Y 24Y 34 ⎥.⎦Y M−1,M∈ R M(M−1)/2 (63)Like svec (47), dvec is an isometric isomorphism on the symmetric hollowsubspace.