v2009.01.01 - Convex Optimization

2.5. SUBSPACE REPRESENTATIONS 81
If the two planes are independent (meaning any line in one is linearly
independent [ ] of any line from the other), they will intersect at a point because
A1
then is invertible;
A 2
{ ∣ [ ] [ ]}
∣∣∣
A 1 ∩ A 2 = x∈ R 4 A1 b1
x =
(142)
A 2 b 2
2.5.1.2.2 Exercise. Linear program.
Minimize a hyperplane over affine set A in the nonnegative orthant
minimize c T x
x
subject to Ax = b
x ≽ 0
(143)
where A = {x | Ax = b}. Two cases of interest are drawn in Figure 26.
Graphically illustrate and explain optimal solutions indicated in the caption.
Why is α ⋆ negative in both cases? Is there solution on the vertical axis?
2.5.2 Intersection of subspaces
The intersection of nullspaces associated with two matrices A∈ R m×n and
B ∈ R k×n can be expressed most simply as
([ ]) [ ]
A ∆ A
N(A) ∩ N(B) = N = {x∈ R n | x = 0} (144)
B
B
the nullspace of their rowwise concatenation.
Suppose the columns of a matrix Z constitute a basis for N(A) while the
columns of a matrix W constitute a basis for N(BZ). Then [134,12.4.2]
N(A) ∩ N(B) = R(ZW) (145)
If each basis is orthonormal, then the columns of ZW constitute an
orthonormal basis for the intersection.
In the particular circumstance A and B are each positive semidefinite
[20,6], or in the circumstance A and B are two linearly independent dyads
(B.1.1), then
N(A) ∩ N(B) = N(A + B),
⎧
⎨
⎩
A,B ∈ S M +
or
A + B = u 1 v T 1 + u 2 v T 2
(l.i.)
(146)