v2009.01.01 - Convex Optimization

2.3. HULLS 57
The affine hull of two distinct points is the unique line through them.
(Figure 17) The affine hull of three noncollinear points in any dimension
is that unique plane containing the points, and so on. The subspace of
symmetric matrices S m is the affine hull of the cone of positive semidefinite
matrices; (2.9)
aff S m + = S m (75)
2.3.1.0.2 Example. Affine hull of rank-1 correlation matrices. [189]
The set of all m ×m rank-1 correlation matrices is defined by all the binary
vectors y in R m (confer5.9.1.0.1)
{yy T ∈ S m + | δ(yy T )=1} (76)
Affine hull of the rank-1 correlation matrices is equal to the set of normalized
symmetric matrices; id est,
aff{yy T ∈ S m + | δ(yy T )=1} = {A∈ S m | δ(A)=1} (77)
2.3.1.0.3 Exercise. Affine hull of correlation matrices.
Prove (77) via definition of affine hull. Find the convex hull instead.
2.3.1.1 Partial order induced by R N + and S M +
Notation a ≽ 0 means vector a belongs to the nonnegative orthant R N + ,
while a ≻ 0 means vector a belongs to the nonnegative orthant’s
interior int R N + , whereas a ≽ b denotes comparison of vector a to vector b
on R N with respect to the nonnegative orthant; id est, a ≽ b means a −b
belongs to the nonnegative orthant, but neither a or b necessarily belongs
to that orthant. With particular respect to the nonnegative orthant,
a ≽ b ⇔ a i ≥ b i ∀i (333). More generally, a ≽ K
b or a ≻ K
b denotes
comparison with respect to pointed closed convex cone K , but equivalence
with entrywise comparison does not hold. (2.7.2.2)
The symbol ≥ is reserved for scalar comparison on the real line R with
respect to the nonnegative real line R + as in a T y ≥ b . Comparison of
matrices with respect to the positive semidefinite cone S M + , like I ≽A ≽ 0
in Example 2.3.2.0.1, is explained in2.9.0.1.