Download A Matrix Handbook for Statisticians (Wiley Series in by George A. F. Seber PDF

By George A. F. Seber

ISBN-10: 0470226781

ISBN-13: 9780470226780

ISBN-10: 0471748692

ISBN-13: 9780471748694

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Extra resources for A Matrix Handbook for Statisticians (Wiley Series in Probability and Statistics)

Example text

S (4 ( 0 , O ) S = 0 (b) IIX + Ylls I llxlls + IlYlls. (c) N = {x E V : llxlls = 0) is a subspace of V . 17. (Schwarz Inequality) Given an inner product space, we have for all x and y (x,Y)2I (X,X)(YlY), or I(X,Y)I I llxll . IlYlll with equality if either x or y is zero or x = ky for some scalar k . We can obtain various inequalities from the above by changing the inner product space (cf. 1). 18. Given an inner product space and unit vectors u, v, and w, then Jm IJl - ((U,W)l2 + J1 - I(W,V)l2.

A basis whose vectors are mutually orthogonal with unit length is called an orthonormal basis. 30). 23. Let V and W be vector subspaces of a vector space U such that V g W . Any orthonormal basis for V can be enlarged t o form an orthonormal basis for W . 16. Let U be a vector space over F with an inner product ( , ) , and let V be a subset or subspace of U . Then the orthogonal complement of V with respect to U is defined to be V' = {x : (x,y)= o for all y E v}. If V and W are two vector subspaces, we say that V I W if ( x , y ) = 0 for all x E V and y E W .

5 ) w2 c w1. ,, = 2P,, (P,, +P,,)+P,, = 2P,,(P,, +P,,)+P,, the Moore-Penrose inverse of B. The above results hold for Q1" if (x,y) = y*x and ' is replaced by Here B+ denotes *. 22. (Centering) Let a = ( a i ) be an n x 1 real vector, and let ? =i Cy=lail.. We say that the a is centered when we transform ai to bi = ai - Ti. If we have the n x p matrix A = (al,az,.. a,)' = (a(1),a(2), . . ,a(,)) and a = n-1 C= ;"= ,i, then we say that A is row centered if we transform it t o the matrix B = (a1 - a,a2 - a,..

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