v2009.01.01 - Convex Optimization

E.1. IDEMPOTENT MATRICES 647
When the domain is restricted to range of P , say x=Uξ for ξ ∈ R k , then
x = Px = UQ T Uξ = Uξ and expansion is unique because the eigenvectors
are linearly independent. Otherwise, any component of x in N(P)= N(Q T )
will be annihilated. Direction of nonorthogonal projection is orthogonal to
R(Q) ⇔ Q T U =I ; id est, for Px∈ R(U)
Px − x ⊥ R(Q) in R m (1774)
E.1.1.0.1 Example. Illustration of nonorthogonal projector.
Figure 138 shows cone(U) , the conic hull of the columns of
⎡ ⎤
1 1
U = ⎣−0.5 0.3 ⎦ (1775)
0 0
from nonorthogonal projector P = UQ T . Matrix U has a limitless number
of left inverses because N(U T ) is nontrivial. Similarly depicted is left inverse
Q T from (1762)
⎡
Q = U †T + ZB T = ⎣
⎡
= ⎣
0.3750 0.6250
−1.2500 1.2500
0 0
0.3750 0.6250
−1.2500 1.2500
0.5000 0.5000
⎤
⎡
⎦ + ⎣
⎤
⎦
0
0
1
⎤
⎦[0.5 0.5]
(1776)
where Z ∈ N(U T ) and matrix B is selected arbitrarily; id est, Q T U = I
because U is full-rank.
Direction of projection on R(U) is orthogonal to R(Q). Any point along
line T in the figure, for example, will have the same projection. Were matrix
Z instead equal to 0, then cone(Q) would become the relative dual to
cone(U) (sharing the same affine hull;2.13.8, confer Figure 49(a)). In that
case, projection Px = UU † x of x on R(U) becomes orthogonal projection
(and unique minimum-distance).
E.1.2
Idempotence summary
Nonorthogonal subspace-projector P is a linear operator defined by
idempotence or biorthogonal decomposition (1765), but characterized not
by symmetry nor positive semidefiniteness nor nonexpansivity (1792).