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Unformatted text preview: zation (Exercise 3.9.8, p. 140). If orthogonal reduction (p. 341) is used to produce a unitary matrix P = P1 P2 and an upper-trapezoidal matrix T = T1 0 such that PA = T, where P1 is r × m and T1 contains the nonzero rows, then M = P∗ T1 is a full-rank factorization. If 1 M=U Copyright c 2000 SIAM D 0 0 0 V∗ = (U1 | U2 ) D 0 0 0 ∗ V1 ∗ V2 ∗ = U1 DV1 (7.10.38) Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 634 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 is the singular value decomposition (5.12.2) on p. 412 (a URV factorization ∗ ∗ (p. 407) could also be used), then M = U1 (DV1 ) = (U1 D)V1 are full-rank factorizations. Projectors, in general, and limiting projectors, in particular, are nicely described in terms of full-rank factorizations. It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Projectors If Mn×n = Bn×r Cr×n is any full-rank factorization as described in (7.10.37), and if R (M) and N (M) are complementary subspaces of C n , then the projector onto R (M) along N (M) is given by P = B(CB)−1 C or D E (7.10.39) ∗ ∗ P = U1 (V1 U1 )−1 V1 when (7.10.38) is used. (7.10.40) T H If A is convergent or summable to G as described in (7.10.34) and (7.10.36), and if I − A = BC is a full-rank factorization, then G = I − B(CB)−1 C or (7.10.41) IG R ∗ ∗ G = I − U1 (V1 U1 )−1 V1 when (7.10.38) is used. (7.10.42) Note: Formulas (7.10.39) and (7.10.40) are extensions of (5.13.3) on p. 430. Proof. Y P It’s always true (Exercise 4.5.12, p. 220) that R (Xm×n Yn×p ) = R (X) when rank (Y) = n, (7.10.43) N (Xm×n Yn×p ) = N (Y) when rank (X) = n. If Mn×n = Bn×r Cr×n is a full-rank factorization, and if R (M) and N (M) are complementary subspaces of C N , then rank (M) = rank M2 (Exercise 5.10.12, p. 402), so combining this with the first part of (7.10.43) produces O C r = rank (BC) = rank (BCBC) = rank (CB)r×r =⇒ (CB)−1 exists. P = B(CB)−1 C is a projector because P2 = P (rec...
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