Download e-book for kindle: A Matrix Handbook for Statisticians (Wiley Series in by George A. F. Seber

By George A. F. Seber

ISBN-10: 0470226781

ISBN-13: 9780470226780

ISBN-10: 0471748692

ISBN-13: 9780471748694

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

Example text

15. 17). Then, since ( x , y ) ( y , x ) is real, + b,Y) + (Y,X) I I(X>Y)+ (Y,X)I I I(X,Y)I + I(Y,X)I + I I1xJI. YIO + + which proves (e). We obtain (a) by writing IIx y1I2 = ( x y , x y ) and using (e); the rest are straightforward. See also Rao and Rao [1998: 541. 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.

22a. Searle [1982: 63, real case]. 2213. 21b) and trace[(XAY)(XAY)*]= trace(X*XAYY*A*)= 0. 3 . Magnus and Neudecker [1999: 341. 23. Let A and B be any any weak inverses. 11c), taking transposes and noting that rankC = rankC' for any C, we have the following results. + rankB rank A + rank B dim[C(A)n C(B)] = rankA dim[C(A') n C(B')] = (b) 0 5 rank M 5 min{c, d } . Hence c = 0 or d =0 - rank(A, B) - rank (= c say), (= d say). implies M = 0. 41 MATRIX SUMS rank ( O A B A A 0 B O B ) = rank(A,B) + r a n k = rank A (t)+ rank M + rank B + rank(A + B) From the above we have the following: + rank (t)+ rank M = rank A + rank B + rank(A + B).

Harville [2001: 27, exercise 11 and Marsaglia and Styan [1974a: theorem 21. 21. Searle [1982: 62-63, real case]; (a) implies (b). 22a. Searle [1982: 63, real case]. 2213. 21b) and trace[(XAY)(XAY)*]= trace(X*XAYY*A*)= 0. 3 . Magnus and Neudecker [1999: 341. 23. Let A and B be any any weak inverses. 11c), taking transposes and noting that rankC = rankC' for any C, we have the following results. + rankB rank A + rank B dim[C(A)n C(B)] = rankA dim[C(A') n C(B')] = (b) 0 5 rank M 5 min{c, d } . Hence c = 0 or d =0 - rank(A, B) - rank (= c say), (= d say).

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A Matrix Handbook for Statisticians (Wiley Series in Probability and Statistics) by George A. F. Seber


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