v2010.10.26 - Convex Optimization

E.3. SYMMETRIC IDEMPOTENT MATRICES 713E.3.5Unifying trait of all projectors: directionRelation (1922) 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) :n∑x = UU T x = u i u T i x (1923)a sum of one-dimensional orthogonal projections (E.6.3), wherei=1U [u 1 · · · u n ] and U T U = I (1924)and where the subspace projector has two expressions, (1922)AA † UU T (1925)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 ;E.3.5.2u 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})orthogonal projector, biorthogonal decomposition(1926)We get a similar result for the biorthogonal expansion of x∈ R(A). DefineA [a 1 a 2 · · · a n ] ∈ R m×n (1927)and the rows of the pseudoinverse E.8⎡ ⎤a ∗T1A † a∗T⎢⎣2 ⎥. ⎦ ∈ Rn×m (1928)a ∗TnE.8 Notation * in this context connotes extreme direction of a dual cone; e.g., (407) orExample E.5.0.0.3.