v2010.10.26 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 57In the ambient space of real matrices, the antisymmetric matrix subspacecan be described{ }1S M⊥ =2 (A −AT ) | A∈ R M×M ⊂ R M×M (55)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 of symmetric matrix subspaceWhen a matrix is symmetric in S M , we may still employ the vectorizationtransformation (37) 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 take symmetric vectorization [215,2.2.1]⎡ ⎤√2Y12Y 11Y 22√2Y13svec Y √2Y23∈ R M(M+1)/2 (56)⎢ Y 33 ⎥⎣ ⎦.Y MMwhere all entries off the main diagonal have been scaled. Now for Z ∈ S M〈Y , Z〉 tr(Y T Z) = vec(Y ) T vec Z = svec(Y ) T svec Z (57)Then because the metrics become equivalent, for X ∈ S M‖ svec X − svec Y ‖ 2 = ‖X − Y ‖ F (58)and because symmetric vectorization (56) is a linear bijective mapping, thensvec is an isometric isomorphism of the symmetric matrix subspace. In other