v2009.01.01 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 51
All antisymmetric matrices are hollow by definition (have 0 main
diagonal). Any square matrix A ∈ R M×M can be written as the sum of
its symmetric and antisymmetric parts: respectively,
A = 1 2 (A +AT ) + 1 2 (A −AT ) (46)
The symmetric part is orthogonal in R M2 to the antisymmetric part; videlicet,
tr ( (A T + A)(A −A T ) ) = 0 (47)
In the ambient space of real matrices, the antisymmetric matrix subspace
can be described
{ }
S M⊥ =
∆ 1
2 (A −AT ) | A∈ R M×M ⊂ R M×M (48)
because any matrix in S M is orthogonal to any matrix in S M⊥ . Further
confined to the ambient subspace of symmetric matrices, because of
antisymmetry, S M⊥ would become trivial.
2.2.2.1 Isomorphism of symmetric matrix subspace
When a matrix is symmetric in S M , we may still employ the vectorization
transformation (30) to R M2 ; vec , an isometric isomorphism. We might
instead choose to realize in the lower-dimensional subspace R M(M+1)/2 by
ignoring redundant entries (below the main diagonal) during transformation.
Such a realization would remain isomorphic but not isometric. Lack of
isometry is a spatial distortion due now to disparity in metric between R M2
and R M(M+1)/2 . To realize isometrically in R M(M+1)/2 , we must make a
correction: For Y = [Y ij ]∈ S M we introduce the symmetric vectorization
⎡ ⎤
√2Y12
Y 11
Y 22
svec Y =
∆ √2Y13
√2Y23
∈ R M(M+1)/2 (49)
⎢ Y 33 ⎥
⎣ ⎦
.
Y MM