By J. P. Bickel, N. El Karoui, M. Yor, P. L. Hennequin
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Extra resources for Ecole d'Ete de Probabilites de Saint-Flour IX - 1979
16. Rao and Rao [1998: 771. 17. , Schott [2005: 361 and Ben-Israel and Greville [2003: 71). The inequality also holds for quasi-inner (semi-inner) products (Harville [1997: 2551). 18. Zhang [1999: 1551. 20. 13. A function f defined on a vector space V over a field F and taking values in F is said to be a linear functional if f(QlX1 + m x 2 ) = Olf (x1) + a z f ( x 2 ) for every XI, x2 E V and every cq,a2 E IF. 71. 21. (Riesz) Let V be an an inner product space with inner product (,), and let f be a linear functional on V .
Tp),where the columns ti of T form an orthonormal basis for V . Then PV = TT*, and the projection of v onto V is v1 = TT*v = C;=l(tfv)tz. (d) If V = C(X), then PV = X(X*X)-X* = XX+, where (X*X)- is a weak inverse of X*X and Xf is the Moore-Penrose inverse of X. When the columns of X are linearly independent, PV= X(X*X)-lX*. (e) Let V = N ( A ) , the null space of A. 37), = I, - A*(AA*)-A. (f) If F” = R”, then the previous results hold by replacing * by ’ and replacing Hermitian by real symmetric.
21, and Seber and Lee [2003: 2031). 53. Let w1 and w2 be vector subspaces of (a) P = P,, case P,, (b) If + P,, + P,, R" with inner product (x,y) = x'y. is an orthogonal projector if and only if w = w1@ w2. w1 Iw2, in which = P,, where w1 = C(A) and w2 = C(B) in (a),then w 1 @ w2 = C(A,B). (c) The following statements are equivalent: (1) P,, - P,, is an orthogonal projection matrix. (2) llPwlx1122 IIPw2x112for all x E R". (3) p,,p,, = p,,. (4)p,,p,, = p,,. ( 5 ) w2 c w1. ,, = 2P,, (P,, +P,,)+P,, = 2P,,(P,, +P,,)+P,, the Moore-Penrose inverse of B.
Ecole d'Ete de Probabilites de Saint-Flour IX - 1979 by J. P. Bickel, N. El Karoui, M. Yor, P. L. Hennequin