v2009.01.01 - Convex Optimization

54 CHAPTER 2. CONVEX GEOMETRY
Any matrix A∈ R M×M can be written as the sum of its symmetric hollow
and antisymmetric antihollow parts: respectively,
( ) ( 1 1
A =
2 (A +AT ) − δ 2 (A) +
2 (A −AT ) + δ (A))
2 (62)
The symmetric hollow part is orthogonal in R M2 to the antisymmetric
antihollow part; videlicet,
))
1 1
tr((
2 (A +AT ) − δ (A))( 2 2 (A −AT ) + δ 2 (A) = 0 (63)
In the ambient space of real matrices, the antisymmetric antihollow subspace
is described
{ }
∆ 1
=
2 (A −AT ) + δ 2 (A) | A∈ R M×M ⊆ R M×M (64)
S M⊥
h
because any matrix in S M h is orthogonal to any matrix in S M⊥
h . Yet in
the ambient space of symmetric matrices S M , the antihollow subspace is
nontrivial;
S M⊥
h
∆
= { δ 2 (A) | A∈ S M} = { δ(u) | u∈ R M} ⊆ S M (65)
In anticipation of their utility with Euclidean distance matrices (EDMs)
in5, for symmetric hollow matrices we introduce the linear bijective
vectorization dvec that is the natural analogue to symmetric matrix
vectorization svec (49): for Y = [Y ij ]∈ S M h
⎡
dvecY = ∆ √ 2
⎢
⎣
⎤
Y 12
Y 13
Y 23
Y 14
Y 24
Y 34 ⎥
.
⎦
Y M−1,M
∈ R M(M−1)/2 (66)
Like svec (49), dvec is an isometric isomorphism on the symmetric hollow
subspace.