v2010.10.26 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 59Operating on a vector, linear operator δ naturally returns a diagonal matrix;δ(δ(A)) is a diagonal matrix. Operating recursively on a vector Λ∈ R N ordiagonal matrix Λ∈ S N , operator δ(δ(Λ)) returns Λ itself;δ 2 (Λ) ≡ δ(δ(Λ)) = Λ (1417)The subspace R M×Mh(63) comprising (real) symmetric hollow matrices isisomorphic with subspace R M(M−1)/2 ; its orthogonal complement isR M×M⊥h { A∈ R M×M | A=−A T + 2δ 2 (A) } ⊆ R M×M (64)the subspace of antisymmetric antihollow matrices in R M×M ; id est,R M×Mh⊕ R M×M⊥h= R M×M (65)Yet defined instead as a proper subspace of ambient S MS M h { A∈ S M | δ(A) = 0 } ≡ R M×Mh⊂ S M (66)the orthogonal complement S M⊥h of symmetric hollow subspace S M h ,S M⊥h { A∈ S M | A=δ 2 (A) } ⊆ S M (67)called symmetric antihollow subspace, is simply the subspace of diagonalmatrices; id est,S M h ⊕ S M⊥h = S M (68)△Any matrix A∈ R M×M can be written as a sum of its symmetric hollowand antisymmetric antihollow parts: respectively,( ) ( 1 1A =2 (A +AT ) − δ 2 (A) +2 (A −AT ) + δ (A))2 (69)The symmetric hollow part is orthogonal to the antisymmetric antihollowpart in R M2 ; videlicet,))1 1tr((2 (A +AT ) − δ (A))( 2 2 (A −AT ) + δ 2 (A) = 0 (70)