v2009.01.01 - Convex Optimization

78 CHAPTER 2. CONVEX GEOMETRY
2.4.3 Angle between hyperspaces
Given halfspace-descriptions, the dihedral angle between hyperplanes and
halfspaces is defined as the angle between their defining normals. Given
normals a and b respectively describing ∂H a and ∂H b , for example
( )
(∂H a , ∂H b ) = ∆ 〈a , b〉
arccos radians (126)
‖a‖ ‖b‖
2.5 Subspace representations
There are two common forms of expression for Euclidean subspaces, both
coming from elementary linear algebra: range form and nullspace form;
a.k.a, vertex-description and halfspace-description respectively.
The fundamental vector subspaces associated with a matrix A∈ R m×n
[287,3.1] are ordinarily related
and of dimension:
with complementarity
R(A T ) ⊥ N(A), N(A T ) ⊥ R(A) (127)
R(A T ) ⊕ N(A) = R n , N(A T ) ⊕ R(A) = R m (128)
dim R(A T ) = dim R(A) = rankA ≤ min{m,n} (129)
dim N(A) = n − rankA , dim N(A T ) = m − rankA (130)
From these four fundamental subspaces, the rowspace and range identify one
form of subspace description (range form or vertex-description (2.3.4))
R(A T ) ∆ = spanA T = {A T y | y ∈ R m } = {x∈ R n | A T y=x , y ∈R(A)} (131)
R(A) ∆ = spanA = {Ax | x∈ R n } = {y ∈ R m | Ax=y , x∈R(A T )} (132)
while the nullspaces identify the second common form (nullspace form or
halfspace-description (104))
N(A) ∆ = {x∈ R n | Ax=0} (133)
N(A T ) ∆ = {y ∈ R m | A T y=0} (134)