v2007.09.17 - Convex Optimization

E.3. SYMMETRIC IDEMPOTENT MATRICES 593E.3.2.0.1 Theorem. Symmetric rank/trace. (confer (1673) (1260))P T = P , P 2 = P⇔rankP = trP = ‖P ‖ 2 F and rank(I − P ) = tr(I − P ) = ‖I − P ‖ 2 F(1688)⋄Proof. We take as given Theorem E.2.0.0.1 establishing idempotence.We have left only to show trP =‖P ‖ 2 F ⇒ P T = P , established in [301,7.1].E.3.3Summary, symmetric idempotentIn summary, orthogonal projector P is a linear operator defined[148,A.3.1] by idempotence and symmetry, and characterized bypositive semidefiniteness and nonexpansivity. The algebraic complement(E.2) to R(P ) becomes the orthogonal complement R(I − P ) ; id est,R(P ) ⊥ R(I − P ).E.3.4Orthonormal decompositionWhen Z = 0 in the general nonorthogonal projector A(A † + BZ T ) (1654),an orthogonal projector results (for any matrix A) characterized principallyby idempotence and symmetry. Any real orthogonal projector may, infact, be represented by an orthonormal decomposition such as (1680).[152,1, prob.42]To verify that assertion for the four fundamental subspaces (1678),we need only to express A by subcompact singular value decomposition(A.6.2): From pseudoinverse (1363) of A = UΣQ T ∈ R m×nAA † = UΣΣ † U T = UU T ,I − AA † = I − UU T = U ⊥ U ⊥T ,A † A = QΣ † ΣQ T = QQ TI − A † A = I − QQ T = Q ⊥ Q ⊥T(1689)where U ⊥ ∈ R m×m−rank A holds columnar an orthonormal basis for theorthogonal complement of R(U) , and likewise for Q ⊥ ∈ R n×n−rank A .Existence of an orthonormal decomposition is sufficient to establishidempotence and symmetry of an orthogonal projector (1680).