v2009.01.01 - Convex Optimization

644 APPENDIX E. PROJECTION
When matrix A in (1762) is skinny full-rank (A † A = I) or has orthonormal
columns (A T A = I), either property leads to a biorthogonal characterization
of nonorthogonal projection:
E.1.1
Biorthogonal characterization
Any nonorthogonal projector P 2 =P ∈ R m×m projecting on nontrivial R(U)
can be defined by a biorthogonality condition Q T U =I ; the biorthogonal
decomposition of P being (confer (1762))
where E.3 (B.1.1.1)
and where generally (confer (1791)) E.4
P = UQ T , Q T U = I (1765)
R(P )= R(U) , N(P )= N(Q T ) (1766)
P T ≠ P , P † ≠ P , ‖P ‖ 2 ≠ 1, P 0 (1767)
and P is not nonexpansive (1792).
(⇐) To verify assertion (1765) we observe: because idempotent matrices
are diagonalizable (A.5), [176,3.3, prob.3] they must have the form (1438)
P = SΦS −1 =
m∑
φ i s i wi T =
i=1
k∑
≤ m
i=1
s i w T i (1768)
that is a sum of k = rankP independent projector dyads (idempotent
dyads,B.1.1,E.6.2.1) where φ i ∈ {0, 1} are the eigenvalues of P
[344,4.1, thm.4.1] in diagonal matrix Φ∈ R m×m arranged in nonincreasing
E.3 Proof. Obviously, R(P ) ⊆ R(U). Because Q T U = I
R(P ) = {UQ T x | x ∈ R m }
⊇ {UQ T Uy | y ∈ R k } = R(U)
E.4 Orthonormal decomposition (1788) (conferE.3.4) is a special case of biorthogonal
decomposition (1765) characterized by (1791). So, these characteristics (1767) are not
necessary conditions for biorthogonality.