v2007.09.13 - Convex Optimization

E.3. SYMMETRIC IDEMPOTENT MATRICES 593under the biorthogonality condition A † A=I . In the biorthogonal expansion(2.13.8)n∑x = AA † x = a i a ∗Ti x (1691)the 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 ;i=1a ∗Tj (a j a ∗Tj x − x) = 0a j a ∗Tj x − x = a j a ∗Tj x − AA † x ∈ R({a i |i=1... n, i≠j})(1692)E.3.5.3nonorthogonal projector, biorthogonal decompositionBecause the result inE.3.5.2 is independent of matrix symmetryAA † =(AA † ) T , we must get the same result for any nonorthogonal projectorcharacterized by a biorthogonality condition; namely, for nonorthogonalprojector P = UQ T (1652) under biorthogonality condition Q T U = I , inthe biorthogonal expansion of x∈ R(U)x = UQ T x =k∑u i qi T x (1693)i=1whereU = ∆ [ ]u 1 · · · u k ∈ Rm×k⎡ ⎤q T1 (1694)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})(1695)