v2010.10.26 - Convex Optimization
v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization
58 CHAPTER 2. CONVEX GEOMETRYwords, S M is isometrically isomorphic with R M(M+1)/2 in the Euclidean senseunder transformation svec .The set of all symmetric matrices S M forms a proper subspace in R M×M ,so for it there exists a standard orthonormal basis in isometrically isomorphicR M(M+1)/2 ⎧⎨{E ij ∈ S M } =⎩e i e T i ,1 √2(e i e T j + e j e T ii = j = 1... M), 1 ≤ i < j ≤ M⎫⎬⎭(59)where M(M + 1)/2 standard basis matrices E ij are formed from thestandard basis vectors[{ ]1, i = je i =0, i ≠ j , j = 1... M ∈ R M (60)Thus we have a basic orthogonal expansion for Y ∈ S MY =M∑ j∑〈E ij ,Y 〉 E ij (61)j=1 i=1whose coefficients〈E ij ,Y 〉 ={Yii , i = 1... MY ij√2 , 1 ≤ i < j ≤ M(62)correspond to entries of the symmetric vectorization (56).2.2.3 Symmetric hollow subspace2.2.3.0.1 Definition. Hollow subspaces. [352]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 (63)where the main diagonal of A∈ R M×M is denoted (A.1)δ(A) ∈ R M (1415)
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)
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58 CHAPTER 2. CONVEX GEOMETRYwords, S M is isometrically isomorphic with R M(M+1)/2 in the Euclidean senseunder transformation svec .The set of all symmetric matrices S M forms a proper subspace in R M×M ,so for it there exists a standard orthonormal basis in isometrically isomorphicR M(M+1)/2 ⎧⎨{E ij ∈ S M } =⎩e i e T i ,1 √2(e i e T j + e j e T ii = j = 1... M), 1 ≤ i < j ≤ M⎫⎬⎭(59)where M(M + 1)/2 standard basis matrices E ij are formed from thestandard basis vectors[{ ]1, i = je i =0, i ≠ j , j = 1... M ∈ R M (60)Thus we have a basic orthogonal expansion for Y ∈ S MY =M∑ j∑〈E ij ,Y 〉 E ij (61)j=1 i=1whose coefficients〈E ij ,Y 〉 ={Yii , i = 1... MY ij√2 , 1 ≤ i < j ≤ M(62)correspond to entries of the symmetric vectorization (56).2.2.3 Symmetric hollow subspace2.2.3.0.1 Definition. Hollow subspaces. [352]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 (63)where the main diagonal of A∈ R M×M is denoted (A.1)δ(A) ∈ R M (1415)
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