v2007.09.17 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 512.2.3 Symmetric hollow subspace2.2.3.0.1 Definition. Hollow subspaces. [266]Define a subspace of R M×M : the convex set of all (real) symmetric M ×Mmatrices having 0 main diagonal;R M×Mh∆= { A∈ R M×M | A=A T , δ(A) = 0 } ⊂ R M×M (53)where the main diagonal of A∈ R M×M is denoted (A.1)δ(A) ∈ R M (1220)Operating 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 (1222)The subspace R M×Mh(53) comprising (real) symmetric hollow matrices isisomorphic with subspace R M(M−1)/2 . The orthogonal complement of R M×MhisR M×M⊥h∆= { A∈ R M×M | A=−A T + 2δ 2 (A) } ⊆ R M×M (54)the subspace of antisymmetric antihollow matrices in R M×M ; id est,R M×Mh⊕ R M×M⊥h= R M×M (55)Yet defined instead as a proper subspace of S MS M h∆= { A∈ S M | δ(A) = 0 } ⊂ S M (56)the orthogonal complement S M⊥h of S M h in ambient S MS M⊥h∆= { A∈ S M | A=δ 2 (A) } ⊆ S M (57)is simply the subspace of diagonal matrices; id est,S M h ⊕ S M⊥h = S M (58)△