v2007.09.17 - Convex Optimization

E.2. I − P , PROJECTION ON ALGEBRAIC COMPLEMENT 589E.1.2Idempotence summaryNonorthogonal subspace-projector P is a linear operator defined byidempotence or biorthogonal decomposition (1657), but characterized notby symmetry nor positive semidefiniteness nor nonexpansivity (1684).E.2 I−P , Projection on algebraic complementIt follows from the diagonalizability of idempotent matrices that I − P mustalso be a projection matrix because it too is idempotent, and because it maybe expressedm∑I − P = S(I − Φ)S −1 = (1 − φ i )s i wi T (1669)where (1 − φ i ) ∈ {1, 0} are the eigenvalues of I − P (1256) whoseeigenvectors s i ,w i are identical to those of P in (1660). A consequence ofthat complementary relationship of eigenvalues is the fact, [259,2] [255,2]for subspace projector P = P 2 ∈ R m×mR(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 )(1670)that is easy to see from (1660) and (1669). Idempotent I −P thereforeprojects vectors on its range, N(P ). Because all eigenvectors of a realidempotent matrix are real and independent, the algebraic complement ofR(P ) [166,3.3] 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 mi=1(1671)because R(P ) ⊕ R(I −P )= R m . For idempotent P ∈ R m×m , consequently,rankP + rank(I − P ) = m (1672)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 algebraiccomplement is the orthogonal complement.