v2010.10.26 - Convex Optimization

710 APPENDIX E. PROJECTIONWe get P =AA † so this projection matrix must be symmetric. Then forany matrix A∈ R m×n , symmetric idempotent P projects a given vector xin R m orthogonally on R(A). Under either condition (1907) or (1908), theprojection Px is unique minimum-distance; for subspaces, perpendicularityand minimum-distance conditions are equivalent.E.3.1Four subspacesWe summarize the orthogonal projectors projecting on the four fundamentalsubspaces: for A∈ R m×nA † A : R n on ( R(A † A) = R(A T ) )AA † : R m on ( R(AA † ) = R(A) )I −A † A : R n on ( R(I −A † A) = N(A) )I −AA † : R m on ( R(I −AA † )= N(A T ) ) (1910)A basis for each fundamental subspace comprises the linearly independentcolumn vectors from its associated symmetric projection matrix:basis R(A T ) ⊆ A † A ⊆ R(A T )basis R(A) ⊆ AA † ⊆ R(A)basis N(A) ⊆ I −A † A ⊆ N(A)basis N(A T ) ⊆ I −AA † ⊆ N(A T )(1911)For completeness: E.7 (1902)N(A † A) = N(A)N(AA † ) = N(A T )N(I −A † A) = R(A T )N(I −AA † ) = R(A)(1912)E.3.2Orthogonal characterizationAny symmetric projector P 2 =P ∈ S m projecting on nontrivial R(Q) canbe defined by the orthonormality condition Q T Q = I . When skinny matrixE.7 Proof is by singular value decomposition (A.6.2): N(A † A)⊆ N(A) is obvious.Conversely, suppose A † Ax=0. Then x T A † Ax=x T QQ T x=‖Q T x‖ 2 =0 where A=UΣQ Tis the subcompact singular value decomposition. Because R(Q)= R(A T ) , then x∈N(A)which implies N(A † A)⊇ N(A). ∴ N(A † A)= N(A).