v2009.01.01 - Convex Optimization

E.7. ON VECTORIZED MATRICES OF HIGHER RANK 671
E.7.2.0.2 Example. PXP redux & N(V).
Suppose we define a subspace of m ×n matrices, each elemental matrix
having columns constituting a list whose geometric center (5.5.1.0.1) is the
origin in R m :
R m×n
c
∆
= {Y ∈ R m×n | Y 1 = 0}
= {Y ∈ R m×n | N(Y ) ⊇ 1} = {Y ∈ R m×n | R(Y T ) ⊆ N(1 T )}
= {XV | X ∈ R m×n } ⊂ R m×n (1873)
the nonsymmetric geometric center subspace. Further suppose V ∈ S n is
a projection matrix having N(V )= R(1) and R(V ) = N(1 T ). Then linear
mapping T(X)=XV is the orthogonal projection of any X ∈ R m×n on R m×n
c
in the Euclidean (vectorization) sense because V is symmetric, N(XV )⊇1,
and R(VX T )⊆ N(1 T ).
Now suppose we define a subspace of symmetric n ×n matrices each of
whose columns constitute a list having the origin in R n as geometric center,
S n c
∆
= {Y ∈ S n | Y 1 = 0}
= {Y ∈ S n | N(Y ) ⊇ 1} = {Y ∈ S n | R(Y ) ⊆ N(1 T )}
(1874)
the geometric center subspace. Further suppose V ∈ S n is a projection
matrix, the same as before. Then V XV is the orthogonal projection of
any X ∈ S n on S n c in the Euclidean sense (1870) because V is symmetric,
V XV 1=0, and R(V XV )⊆ N(1 T ). Two-sided projection is necessary only
to remain in the ambient symmetric matrix subspace. Then
S n c = {V XV | X ∈ S n } ⊂ S n (1875)
has dim S n c = n(n−1)/2 in isomorphic R n(n+1)/2 . We find its orthogonal
complement as the aggregate of all negative directions of orthogonal
projection on S n c : the translation-invariant subspace (5.5.1.1)
S n⊥
c
∆
= {X − V XV | X ∈ S n } ⊂ S n
= {u1 T + 1u T | u∈ R n }
(1876)
characterized by the doublet u1 T + 1u T (B.2). E.13 Defining the
E.13 Proof.
{X − V X V | X ∈ S n } = {X − (I − 1 n 11T )X(I − 11 T 1 n ) | X ∈ Sn }
= { 1 n 11T X + X11 T 1 n − 1 n 11T X11 T 1 n | X ∈ Sn }