By George A. F. Seber
This publication emphasizes computational facts and algorithms and comprises a variety of references to either the idea at the back of the equipment and the functions of the tools. each one bankruptcy contains 4 elements: a definition by means of a listing of effects, a brief checklist of references to similar issues within the e-book (since a few overlap is unavoidable), a number of references to proofs, and references to functions. issues contain unique matrices, non-negative matrices, targeted items and operators, Jacobians, partitioned and patterned matrices, matrix approximation, matrix optimization, a number of integrals and multivariate distributions, linear and quadratic types, and so on.
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Extra resources for A Matrix Handbook for Statisticians
B) rank(AY) = rankA implies rank(KAY) = rank(KA) for every K. 18. Let A , B, and C be conformable matrices, and let (AB)- and (BC)- be any weak inverses. Then: (4 rank( lc ) + rank(BC) + rankL = rank(AB) = r a n k B + rank(ABC), where L = [I - BC(BC)-]B[I - (AB)-(AB)]. (b) (Frobenius Inequality) From (a) we have rank(ABC) 2 rank(AB) + rank(BC) - rankB, with equality if and only if L = 0. 19. Let V be a non-negative definite n x n matrix, and let X be an n x p matrix. Then the following statements are equivalent.
Azx2,y) = (~1x1 By (a), (S'-)'- + + 2 . If x belongs to the left-hand side (LHS), then ( x , s t) = ( x , s ) (x,t ) = 0 for all s E S and all t E T . Setting s = 0, then (x,t) = 0; similarly, (x,s) = 0 and L H S R H S . The argument reverses. 26. Rao and Rao [1998: 62-63]. 27a-b. Harville [1997: 1721. 2 . Harville [2001: 162, exercise 31 and Rao and Bhimasankaram [2000: 2671. 28a(i). 26d) with 24 = W. 28a(ii). If x E R H S , then x = y z where y E V & W and z E W so that x E W and R H S 2 LHS.
A set of vectors independent if EL==, that are not linearly independent are said to be linearly dependent. For further properties of linearly independent sets see Rao and Bhimasankaram [2000: chapter 11. The term “vector” here and in the following definitions is quite general and simply refers to an element of a vector space. For example, it could be an m x n matrix in the vector space of all such matrices; Harville [1997: chapters 3 and 41 takes this approach. 9. A set of vectors vi ( i = 1 , 2 , ..