v2007.09.13 - Convex Optimization

584 APPENDIX E. PROJECTION(⇐) To verify assertion (1652) we observe: because idempotent matricesare diagonalizable (A.5), [149,3.3, prob.3] they must have the form (1334)P = SΦS −1 =m∑φ i s i wi T =i=1k∑≤ mi=1s i w T i (1655)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[298,4.1, thm.4.1] in diagonal matrix Φ∈ R m×m arranged in nonincreasingorder, and where s i ,w i ∈ R m are the right- and left-eigenvectors of P ,respectively, which are independent and real. E.5 ThereforeU ∆ = S(:,1:k) = [ s 1 · · · s k]∈ Rm×k(1656)is the full-rank matrix S ∈ R m×m having m − k columns truncated(corresponding to 0 eigenvalues), while⎡ ⎤Q T = ∆ S −1 (1:k, :) = ⎣w T1.w T k⎦ ∈ R k×m (1657)is matrix S −1 having the corresponding m − k rows truncated. By the0 eigenvalues theorem (A.7.3.0.1), R(U)= R(P ) , R(Q)= R(P T ) , andR(P ) = span {s i | φ i = 1 ∀i}N(P ) = span {s i | φ i = 0 ∀i}R(P T ) = span {w i | φ i = 1 ∀i}N(P T ) = span {w i | φ i = 0 ∀i}(1658)Thus biorthogonality Q T U =I is a necessary condition for idempotence, andso the collection of nonorthogonal projectors projecting on R(U) is the affinesubset P k =UQ T k where Q k = {Q | Q T U = I , Q∈ R m×k }.(⇒) Biorthogonality is a sufficient condition for idempotence;P 2 =k∑s i wiTid est, if the cross-products are annihilated, then P 2 =P .i=1k∑s j wj T = P (1659)j=1E.5 Eigenvectors of a real matrix corresponding to real eigenvalues must be real.(A.5.0.0.1)