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**

Iii) rank(A, B) = r a n k A + rank B if and only if c = 0. (i) rank(A When c = 0, M = 0 and r a n k M = 0 = c so that rank(A (t) + B) = rank (3 + (iv) rank = rankA r a n k B if and only if d = 0. When d = 0. M = 0 and r a n k M = 0 = d so that rank(A (v) rank(A + B) = rank(A, B). + B) = r a n k A + r a n k B if and only if c = d = 0. 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.

5. The following statements are equivalent. (1) B is a quadratic subspace of A. + B)2E B. (3) If A, B E B, then AB + BA E B. (2) If A, B E B, then (A (4) If A E B , then Ak E B for k = 1 , 2 , . .. 6. Let B be a quadratic subspace of A. 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.

For a counter example consider A = (1,i ) ' ( l ,1). 4b. Abadir and Magnus [2005: 801. 5. Ben-Israel and Greville [2003: 261, Marsaglia and Styan [1974a: theorem I ] , and Searle [1982: 1751. 6. 33a) A and B are equivalent to the same diagonal matrix. 14a). 7. Follows from C(AB) = C(AC). 8. By ( l O . 35) we have C(V) = C(R). 7). 2 MATRIX PRODUCTS All the matrices in this section are real or complex. 9. Given conformable matrices A and B, we have the following. (a) rank(BA) = rankA if B has full row rank.

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

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