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By George A. F. Seber

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

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Then: (a) If A , B E B , then ABA E B. (b) Let A E B be fixed and let C = {ABA : B E B } . Then C is a quadratic subspace of B. (c) If A, B, C E B , then ABC + CBA E B. Proofs. 3. 6. Rao and Rao [1998: 434-436, 4401. = 0. 5. As with sets, we define V W t o be the s u m of the two vector subspaces. If V n W = 0 (some authors use { 0 } ) ,we say that V and W are disjoint vector subspaces (Harville [1997] uses the term “essentially disjoint”). Note that this differs from the notion of disjoint sets, namely V n W = 4, which we will not need.

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. 53b. A’B = 0 implies that PAPB= 0. 5 3 ~. Quoted, less generally, by Isotalo et al. [2005a: 611. The proofs are straightforward. For (2), note that for a symmetric idempotent matrix, X’AX= x’A’Ax = llAxll;. 53d. Anderson and Duffin [1969] and Meyer [2000a: 4411. 23. ) on S x S such that: (a) d(x,y ) 2 0 for all x,y E S with equality if and only if x = y ( d is positive definite).

24. 12) that rank(AB - In)5 rank(A - - I, = (A - 1,)B + + I n ) rank(B - In). 2. Suppose that A = C,=lA,, where each matrix is m x n. We say rank A,. 25. Let A and B be nonnull m x n matrices over F with respective ranks s. If any one of the following conditions hold, then they all hold. , rank additivity). T and 42 RANK (2) There exist nonsingular matrices F and G such that A = F ( I,0 0 0 ) G 0 0 and B = F ( O0 I, 0 0 0 0)G. 0 0 The above matrices are partitioned in the same way, and the bordering zero matrices are of appropriate size; some of the latter matrices are absent if A B has full rank.

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