v2010.10.26 - Convex Optimization

E.1. IDEMPOTENT MATRICES 707are linearly independent. Otherwise, any component of x in N(P)= N(Q T )will be annihilated. Direction of nonorthogonal projection is orthogonal toR(Q) ⇔ Q T U =I ; id est, for Px∈ R(U)Px − x ⊥ R(Q) in R m (1898)E.1.1.0.1 Example. Illustration of nonorthogonal projector.Figure 162 shows cone(U) , the conic hull of the columns of⎡ ⎤1 1U = ⎣−0.5 0.3 ⎦ (1899)0 0from nonorthogonal projector P = UQ T . Matrix U has a limitless numberof left inverses because N(U T ) is nontrivial. Similarly depicted is left inverseQ T from (1886)⎡Q = U †T + ZB T = ⎣⎡= ⎣0.3750 0.6250−1.2500 1.25000 00.3750 0.6250−1.2500 1.25000.5000 0.5000⎤⎡⎦ + ⎣⎤⎦001⎤⎦[0.5 0.5](1900)where Z ∈ N(U T ) and matrix B is selected arbitrarily; id est, Q T U = Ibecause U is full-rank.Direction of projection on R(U) is orthogonal to R(Q). Any point alongline T in the figure, for example, will have the same projection. Were matrixZ instead equal to 0, then cone(Q) would become the relative dual tocone(U) (sharing the same affine hull;2.13.8, confer Figure 56a). In thatcase, projection Px = UU † x of x on R(U) becomes orthogonal projection(and unique minimum-distance).E.1.2Idempotence summaryNonorthogonal subspace-projector P is a (convex) linear operator defined byidempotence or biorthogonal decomposition (1889), but characterized not bysymmetry or positive semidefiniteness or nonexpansivity (1917).