v2006.03.09 - Convex Optimization

526 APPENDIX E. PROJECTIONE.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 (1447))where E.3 (B.1.1.1)and where generally (confer (1476)) E.4P = UQ T , Q T U = I (1450)R(P )= R(U) , N(P )= N(Q T ) (1451)P T ≠ P , P † ≠ P , ‖P ‖ 2 ≠ 1, P 0 (1452)and P is not nonexpansive (1477).(⇐) To verify assertion (1450) we observe: because idempotent matricesare diagonalizable (A.5), [125,3.3, prob.3] they must have the form (1144)P = SΦS −1 =m∑φ i s i wi T =i=1k∑≤ mi=1s i w T i (1453)that is a sum of k = rankP independent projector dyads (idempotent dyadB.1.1) where φ i ∈ {0, 1} are the eigenvalues of P [261,4.1, thm.4.1] indiagonal matrix Φ∈ R m×m arranged in nonincreasing order, and wheres i ,w i ∈ R m are the right- and left-eigenvectors of P , respectively, whichare independent and real. E.5 ThereforeU ∆ = S(:,1:k) = [ s 1 · · · s k]∈ Rm×k(1454)E.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 (1473) (conferE.3.4) is a special case of biorthogonaldecomposition (1450) characterized by (1476). So, these characteristics (1452) are notnecessary conditions for biorthogonality.E.5 Eigenvectors of a real matrix corresponding to real eigenvalues must be real.(A.5.0.0.1)