v2009.01.01 - Convex Optimization

648 APPENDIX E. PROJECTION
E.2 I−P , Projection on algebraic complement
It follows from the diagonalizability of idempotent matrices that I − P must
also be a projection matrix because it too is idempotent, and because it may
be expressed
m∑
I − P = S(I − Φ)S −1 = (1 − φ i )s i wi T (1777)
where (1 − φ i ) ∈ {1, 0} are the eigenvalues of I − P (1355) whose
eigenvectors s i ,w i are identical to those of P in (1768). A consequence of
that complementary relationship of eigenvalues is the fact, [298,2] [293,2]
for subspace projector P = P 2 ∈ R m×m
R(P ) = span {s i | φ i = 1 ∀i} = span {s i | (1 − φ i ) = 0 ∀i} = N(I − P )
N(P ) = span {s i | φ i = 0 ∀i} = span {s i | (1 − φ i ) = 1 ∀i} = R(I − P )
R(P T ) = span {w i | φ i = 1 ∀i} = span {w i | (1 − φ i ) = 0 ∀i} = N(I − P T )
N(P T ) = span {w i | φ i = 0 ∀i} = span {w i | (1 − φ i ) = 1 ∀i} = R(I − P T )
(1778)
that is easy to see from (1768) and (1777). Idempotent I −P therefore
projects vectors on its range, N(P ). Because all eigenvectors of a real
idempotent matrix are real and independent, the algebraic complement of
R(P ) [197,3.3] is equivalent to N(P ) ; E.6 id est,
R(P )⊕N(P ) = R(P T )⊕N(P T ) = R(P T )⊕N(P ) = R(P )⊕N(P T ) = R m
i=1
(1779)
because R(P ) ⊕ R(I −P )= R m . For idempotent P ∈ R m×m , consequently,
rankP + rank(I − P ) = m (1780)
E.2.0.0.1 Theorem. Rank/Trace. [344,4.1, prob.9] (confer (1796))
P 2 = P
⇔
rankP = trP and rank(I − P ) = tr(I − P )
(1781)
E.6 The same phenomenon occurs with symmetric (nonidempotent) matrices, for example.
When the summands in A ⊕ B = R m are orthogonal vector spaces, the algebraic
complement is the orthogonal complement.
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