v2007.09.13 - Convex Optimization

592 APPENDIX E. PROJECTIONE.3.5Unifying trait of all projectors: directionRelation (1684) shows: orthogonal projectors simultaneously possessa biorthogonal decomposition (conferE.1.1) (for example, AA † forskinny-or-square A full-rank) and an orthonormal decomposition (UU Twhence Px = UU T x).E.3.5.1orthogonal projector, orthonormal decompositionConsider orthogonal expansion of x∈ R(U) :x = UU T x =n∑u i u T i x (1685)a sum of one-dimensional orthogonal projections (E.6.3), wherei=1U ∆ = [u 1 · · · u n ] and U T U = I (1686)and where the subspace projector has two expressions, (1684)AA † ∆ = UU T (1687)where A ∈ R m×n has rank n . The direction of projection of x on u j forsome j ∈{1... n} , for example, is orthogonal to u j but parallel to a vectorin the span of all the remaining vectors constituting the columns of U ;u T j(u j u T j x − x) = 0u j u T j x − x = u j u T j x − UU T x ∈ R({u i |i=1... n, i≠j})(1688)E.3.5.2orthogonal projector, biorthogonal decompositionWe get a similar result for the biorthogonal expansion of x∈ R(A). Defineand the rows of the pseudoinverse⎡A ∆ = [a 1 a 2 · · · a n ] ∈ R m×n (1689)A † =∆ ⎢⎣a ∗T1a ∗T2 .a ∗Tn⎤⎥⎦ ∈ Rn×m (1690)