v2007.09.13 - Convex Optimization

586 APPENDIX E. PROJECTIONNonorthogonal projection ofbiorthogonal expansion,Px = UQ T x =x on R(P ) has expression like ak∑wi T xs i (1660)When the domain is restricted to the range of P , say x=Uξ forξ ∈ R k , then x = Px = UQ T Uξ = Uξ and expansion is unique because theeigenvectors are linearly independent. Otherwise, any component of xin N(P)= N(Q T ) will be annihilated. The direction of nonorthogonalprojection is orthogonal to R(Q) ⇔ Q T U =I ; id est, for Px∈ R(U)i=1Px − x ⊥ R(Q) in R m (1661)E.1.1.0.1 Example. Illustration of nonorthogonal projector.Figure 117 shows cone(U) , the conic hull of the columns of⎡ ⎤1 1U = ⎣−0.5 0.3 ⎦ (1662)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 (1649)⎡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](1663)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 43(a)) In thatcase, projection Px = UU † x of x on R(U) becomes orthogonal projection(and unique minimum-distance).