v2006.03.09 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 61All antisymmetric matrices are hollow by definition (have 0main-diagonal). Any square matrix A∈ R M×M can be written as the sum ofits symmetric and antisymmetric parts: respectively,A = 1 2 (A +AT ) + 1 2 (A −AT ) (43)The symmetric part is orthogonal in R M2 to the antisymmetric part; videlicet,tr ( (A T + A)(A −A T ) ) = 0 (44)In the ambient space of real matrices, the antisymmetric matrix subspacecan be described{ }S M⊥ =∆ 12 (A −AT ) | A∈ R M×M ⊂ R M×M (45)because any matrix in S M is orthogonal to any matrix in S M⊥ . Furtherconfined to the ambient subspace of symmetric matrices, because ofantisymmetry, S M⊥ would become trivial.2.2.2.1 Isomorphism on symmetric matrix subspaceWhen a matrix is symmetric in S M , we may still employ the vectorizationtransformation (29) to R M2 ; vec , an isometric isomorphism. We mightinstead choose to realize in the lower-dimensional subspace R M(M+1)/2 byignoring redundant entries (below the main diagonal) during transformation.Such a realization would remain isomorphic but not isometric. Lack ofisometry is a spatial distortion due now to disparity in metric between R M2and R M(M+1)/2 . To realize isometrically in R M(M+1)/2 , we must make acorrection: For Y = [Y ij ]∈ S M we introduce the symmetric vectorization⎡ ⎤√2Y12Y 11Y 22svec Y =∆ √2Y13√2Y23∈ R M(M+1)/2 (46)⎢ Y 33 ⎥⎣ ⎦.Y MM