v2009.01.01 - Convex Optimization

E.3. SYMMETRIC IDEMPOTENT MATRICES 651
is an orthogonal projector projecting on R(Q) having, for Px∈ R(Q)
(confer (1774))
Px − x ⊥ R(Q) in R m (1790)
From (1788), orthogonal projector P is obviously positive semidefinite
(A.3.1.0.6); necessarily,
P T = P , P † = P , ‖P ‖ 2 = 1, P ≽ 0 (1791)
and ‖Px‖ = ‖QQ T x‖ = ‖Q T x‖ because ‖Qy‖ = ‖y‖ ∀y ∈ R k . All orthogonal
projectors are therefore nonexpansive because
√
〈Px, x〉 = ‖Px‖ = ‖Q T x‖ ≤ ‖x‖ ∀x∈ R m (1792)
the Bessel inequality, [92] [197] with equality when x∈ R(Q).
From the diagonalization of idempotent matrices (1768) on page 644
P = SΦS T =
m∑
φ i s i s T i =
i=1
k∑
≤ m
i=1
s i s T i (1793)
orthogonal projection of point x on R(P ) has expression like an orthogonal
expansion [92,4.10]
where
Px = QQ T x =
k∑
s T i xs i (1794)
i=1
Q = S(:,1:k) = [ s 1 · · · s k
]
∈ R
m×k
(1795)
and where the s i [sic] are orthonormal eigenvectors of symmetric
idempotent P . When the domain is restricted to range of P , say x=Qξ for
ξ ∈ R k , then x = Px = QQ T Qξ = Qξ and expansion is unique. Otherwise,
any component of x in N(Q T ) will be annihilated.
E.3.2.0.1 Theorem. Symmetric rank/trace. (confer (1781) (1359))
P T = P , P 2 = P
⇔
rankP = trP = ‖P ‖ 2 F and rank(I − P ) = tr(I − P ) = ‖I − P ‖ 2 F
(1796)
⋄