v2010.10.26 - Convex Optimization

E.1. IDEMPOTENT MATRICES 705order, 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 ] ∈ R m×k (1893)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 (1894)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}(1895)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 wiTi=1k∑s j wj T = P (1896)j=1id est, if the cross-products are annihilated, then P 2 =P .Nonorthogonal projection of x on R(P ) has expression like a biorthogonalexpansion,k∑Px = UQ T x = wi T xs i (1897)When the domain is restricted to range of P , say x=Uξ for ξ ∈ R k , thenx = Px = UQ T Uξ = Uξ and expansion is unique because the eigenvectorsE.5 Eigenvectors of a real matrix corresponding to real eigenvalues must be real.(A.5.0.0.1)i=1