v2009.01.01 - Convex Optimization

652 APPENDIX E. PROJECTION
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 [344,7.1].
E.3.3
Summary, symmetric idempotent
In summary, orthogonal projector P is a linear operator defined
[173,A.3.1] by idempotence and symmetry, and characterized by
positive 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.3.4
Orthonormal decomposition
When Z =0 in the general nonorthogonal projector A(A † + BZ T ) (1762),
an orthogonal projector results (for any matrix A) characterized principally
by idempotence and symmetry. Any real orthogonal projector may, in
fact, be represented by an orthonormal decomposition such as (1788).
[178,1, prob.42]
To verify that assertion for the four fundamental subspaces (1786),
we need only to express A by subcompact singular value decomposition
(A.6.2): From pseudoinverse (1463) of A = UΣQ T ∈ R m×n
AA † = UΣΣ † U T = UU T ,
I − AA † = I − UU T = U ⊥ U ⊥T ,
A † A = QΣ † ΣQ T = QQ T
I − A † A = I − QQ T = Q ⊥ Q ⊥T
(1797)
where U ⊥ ∈ R m×m−rank A holds columnar an orthonormal basis for the
orthogonal complement of R(U) , and likewise for Q ⊥ ∈ R n×n−rank A .
Existence of an orthonormal decomposition is sufficient to establish
idempotence and symmetry of an orthogonal projector (1788).
E.3.5
Unifying trait of all projectors: direction
Relation (1797) shows: orthogonal projectors simultaneously possess
a biorthogonal decomposition (conferE.1.1) (for example, AA † for
skinny-or-square A full-rank) and an orthonormal decomposition (UU T
whence Px = UU T x).