v2009.01.01 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 53
2.2.3 Symmetric hollow subspace
2.2.3.0.1 Definition. Hollow subspaces. [305]
Define a subspace of R M×M : the convex set of all (real) symmetric M ×M
matrices having 0 main diagonal;
R M×M
h
∆
= { A∈ R M×M | A=A T , δ(A) = 0 } ⊂ R M×M (56)
where the main diagonal of A∈ R M×M is denoted (A.1)
δ(A) ∈ R M (1318)
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 or
diagonal matrix Λ∈ S N , operator δ(δ(Λ)) returns Λ itself;
δ 2 (Λ) ≡ δ(δ(Λ)) ∆ = Λ (1320)
The subspace R M×M
h
(56) comprising (real) symmetric hollow matrices is
isomorphic with subspace R M(M−1)/2 . The orthogonal complement of R M×M
h
is
R M×M⊥
h
∆
= { A∈ R M×M | A=−A T + 2δ 2 (A) } ⊆ R M×M (57)
the subspace of antisymmetric antihollow matrices in R M×M ; id est,
R M×M
h
⊕ R M×M⊥
h
= R M×M (58)
Yet defined instead as a proper subspace of S M
S M h
∆
= { A∈ S M | δ(A) = 0 } ⊂ S M (59)
the orthogonal complement S M⊥
h of S M h in ambient S M
S M⊥
h
∆
= { A∈ S M | A=δ 2 (A) } ⊆ S M (60)
is simply the subspace of diagonal matrices; id est,
S M h ⊕ S M⊥
h = S M (61)
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