By Paul Lévy
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50b-d. 44). 51a-d(i). Seber and Lee [2003: Appendix B3, 477-478, real case] and Seber [1984: Appendix B3, 535, real case]. 51d(ii). If x E C(X1) = w , then Plx = x, Xh(In - P1)x = 0, and x E N(Xk(1, - P I ) ) . Conversely, if x = Xlal X2a2 E R and 0 = Xk(1, P1)x = XL(1, - P1)X*a2 (since PIX1 = XI), then a 2 = 0 (by (i)) and x E C(X,). 52. 21. 53a. P is clearly symmetric and idempotent if and only P,,P,, = -P,,Pw, . Multiplying on the left by P,, shows that P,,P,, is symmetric and therefore P,,Pw, = 0.
A) Let P v be the orthogonal projection matrix that projects onto V . Then P$ = P v and APv is Hermitian, that is, APv = PGA. (Note that P v is generally not Hermitian. 42). Also PGA(In - P v ) = APV(1, - P v ) = 0. 22 VECTORS, VECTOR SPACES, AND CONVEXITY (c) Let V = C(X). Then Pv = X(X*AX)-X*A, which is unique for any weak inverse (X*AX)- and therefore invariant. Also P V l = I, - P v . (d) If V = C(X),then PvX = X. 47. 46) above, namely ( x , y ) = x’V-’y, where V is positive definite and x , y E R”.
64a-c. Quoted by Rao [1973a: 511. 64d. Schott [2005: 721. 65a. Schott [2005: 711. 6513. Rao [1973a: 511. 65~-d. Rao [1973a: 521 and Schott [2005: 731. 65e. Anderson ,and quoted by Schott [2005: 741. 67. Quoted by Rao [1973a: 531. 69. 71. Magnus and Neudecker [1999: 761. 73. Horn and Johnson [1991: 81 and Zhang [1999: 88-89]. 6 31 COORDINATE GEOMETRY Occasionally one may need some results from coordinate geometry. Some of these are listed below for easy reference. 74. 75. Given the points x1 = (al,bl,cl)' and equation of the line through the points is x2 = y-bl b l - b2 Z--1 ~ Z - U ~ - a1 - a2 x1,x2,.