v2007.11.26 - Convex Optimization

604 APPENDIX E. PROJECTIONE.2.0.0.1 Theorem. Rank/Trace. [303,4.1, prob.9] (confer (1714))P 2 = P⇔rankP = trP and rank(I − P ) = tr(I − P )(1699)⋄E.2.1Universal projector characteristicAlthough projection is not necessarily orthogonal and R(P )̸⊥ R(I − P ) ingeneral, still for any projector P and any x∈ R mPx + (I − P )x = x (1700)must hold where R(I − P ) = N(P ) is the algebraic complement of R(P).The algebraic complement of closed convex cone K , for example, is thenegative dual cone −K ∗ . (1818)E.3 Symmetric idempotent matricesWhen idempotent matrix P is symmetric, P is an orthogonal projector. Inother words, the direction of projection of point x∈ R m on subspace R(P )is orthogonal to R(P ) ; id est, for P 2 =P ∈ S m and projection Px∈ R(P )Px − x ⊥ R(P ) in R m (1701)Perpendicularity is a necessary and sufficient condition for orthogonalprojection on a subspace. [73,4.9]A condition equivalent to (1701) is: Norm of direction x −Px is theinfimum over all nonorthogonal projections of x on R(P ) ; [183,3.3] forP 2 =P ∈ S m , R(P )= R(A) , matrices A,B, Z and positive integer k asdefined for (1680), and given x∈ R m‖x − Px‖ 2 = ‖x − AA † x‖ 2 = infB∈R n×k ‖x − A(A † + BZ T )x‖ 2 (1702)The infimum is attained for R(B)⊆ N(A) over any affine subset ofnonorthogonal projectors (1682) indexed by k .