v2006.03.09 - Convex Optimization

536 APPENDIX E. PROJECTIONthe direction of projection of x on a j for some particular j ∈ {1... n} , forexample, is orthogonal to a † j and parallel to a vector in the span of all theremaining vectors constituting the columns of A ;E.3.4.4a † j (a ja † j x − x) = 0a j a † j x − x = a ja † j x − AA† x ∈ R({a i |i=1... n, i≠j})nonorthogonal projector, biorthogonal decomposition(1490)Because the result inE.3.4.3 is independent of symmetry AA † =(AA † ) T , wemust have the same result for any nonorthogonal projector characterized bya biorthogonality condition; namely, for nonorthogonal projector P = UQ T(1450) under biorthogonality condition Q T U = I , in the biorthogonalexpansion of x∈ R(U)k∑x = UQ T x = u i qi T x (1491)wherei=1U = ∆ [ ]u 1 · · · u k ∈ Rm×k⎡ ⎤q T1 (1492)Q T =∆ ⎣ . ⎦ ∈ R k×mq T kthe direction of projection of x on u j is orthogonal to q j and parallel to avector in the span of the remaining u i :q T j (u j q T j x − x) = 0u j q T j x − x = u j q T j x − UQ T x ∈ R({u i |i=1... k , i≠j})E.4 Algebra of projection on affine subsets(1493)Let P A x denote projection of x on affine subset A ∆ = R + α where R is asubspace and α ∈ A . Then, because R is parallel to A , it holds:P A x = P R+α x = (I − P R )(α) + P R x= P R (x − α) + α(1494)Subspace projector P R is a linear operator (E.1.2), and P R (x + y) = P R xwhenever y ⊥R and P R is an orthogonal projector.