v2010.10.26 - Convex Optimization

704 APPENDIX E. PROJECTIONWhen matrix A in (1886) is skinny full-rank (A † A = I) or has orthonormalcolumns (A T A = I), either property leads to a biorthogonal characterizationof nonorthogonal projection:E.1.1Biorthogonal characterizationAny nonorthogonal projector P 2 =P ∈ R m×m projecting on nontrivial R(U)can be defined by a biorthogonality condition Q T U =I ; the biorthogonaldecomposition of P being (confer (1886))where E.3 (B.1.1.1)and where generally (confer (1916)) E.4P = UQ T , Q T U = I (1889)R(P )= R(U) , N(P )= N(Q T ) (1890)P T ≠ P , P † ≠ P , ‖P ‖ 2 ≠ 1, P 0 (1891)and P is not nonexpansive (1917).(⇐) To verify assertion (1889) we observe: because idempotent matricesare diagonalizable (A.5), [202,3.3 prob.3] they must have the form (1547)P = SΦS −1 =m∑φ i s i wi T =i=1k∑≤ mi=1s i w T i (1892)that is a sum of k = rankP independent projector dyads (idempotentdyads,B.1.1,E.6.2.1) where φ i ∈ {0, 1} are the eigenvalues of P[393,4.1 thm.4.1] in diagonal matrix Φ∈ R m×m arranged in nonincreasingE.3 Proof. Obviously, R(P ) ⊆ R(U). Because Q T U = IR(P ) = {UQ T x | x ∈ R m }⊇ {UQ T Uy | y ∈ R k } = R(U)E.4 Orthonormal decomposition (1913) (conferE.3.4) is a special case of biorthogonaldecomposition (1889) characterized by (1916). So, these characteristics (1891) are notnecessary conditions for biorthogonality.