v2009.01.01 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 71
Recall vector inner-product from2.2, 〈A, Y 〉= tr(A T Y ).
Hyperplanes in R mn may, of course, also be represented using matrix
variables.
∂H = {Y | 〈A, Y 〉 = b} = {Y | 〈A, Y −Y p 〉 = 0} ⊂ R mn (111)
Vector a from Figure 21 is normal to the hyperplane illustrated. Likewise,
nonzero vectorized matrix A is normal to hyperplane ∂H ;
A ⊥ ∂H in R mn (112)
2.4.2.2 Vertex-description of hyperplane
Any hyperplane in R n may be described as the affine hull of a minimal set of
points {x l ∈ R n , l = 1... n} arranged columnar in a matrix X ∈ R n×n (68):
∂H = aff{x l ∈ R n , l = 1... n} ,
dim aff{x l ∀l}=n−1
= aff X , dim aff X = n−1
= x 1 + R{x l − x 1 , l=2... n} , dim R{x l − x 1 , l=2... n}=n−1
= x 1 + R(X − x 1 1 T ) , dim R(X − x 1 1 T ) = n−1
where
R(A) = {Ax | ∀x} (132)
(113)
2.4.2.3 Affine independence, minimal set
For any particular affine set, a minimal set of points constituting its
vertex-description is an affinely independent generating set and vice versa.
Arbitrary given points {x i ∈ R n , i=1... N} are affinely independent
(a.i.) if and only if, over all ζ ∈ R N ζ T 1=1, ζ k = 0 (confer2.1.2)
x i ζ i + · · · + x j ζ j − x k = 0, i≠ · · · ≠j ≠k = 1... N (114)
has no solution ζ ; in words, iff no point from the given set can be expressed
as an affine combination of those remaining. We deduce
l.i. ⇒ a.i. (115)