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Get A Matrix Handbook for Statisticians (Wiley Series in PDF

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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

Also S ( A ) is the smallest subspace of V containing A in the sense that every subspace of V containing A also contains S(A). (c) A is a vector space if and only if A = S ( A ) . (4 S[S(A)I = S ( A ) . (e) If A C B,then S( A ) C S ( B ) . (f) S ( A )u S ( B )c S ( Au B ) . (g) S(A n B ) c S ( A )n w). 8. A set of vectors vi (i = 1,2, . . ,r ) in a vector space are linearly aivi = 0 implies that a1 = a2 = . . = a, = 0. A set of vectors independent if EL==, that are not linearly independent are said to be linearly dependent.

D) dim(V) + dim(V') = dim(U). 27. If V and W are vector subspaces of U , then: (a) V & W if and only if V IW1 (b) V C W if and only if W' (c) (V n W)' = V' + W' & V'. and (V + W)' = V' n WL. 21. 17. Let V and W be vector subspaces of U , a vector space over F, and suppose that V C W. Then the set of all vectors in W that are perpendicular to V form a vector space called the orthogonal complement of V with respect to W , and is denoted by V' n W . 28. Let V (a) E v}. W . Then (i) dim(V' n W) = dim(W) - dim(V).

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.

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


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