# Download A Matrix Handbook for Statisticians (Wiley Series in by George A. F. Seber PDF

By George A. F. Seber

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A real valued function f is convex in an interval I of R if f [ a x+ (1 - a ) y ] I a f ( x )+ ( 1 - a ) f ( y ) , all a such that 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. (1) rank(X’VX) = rankX. (2) rank(X’V+X) = rankX, where V+ is the Moore-Penrose inverse of V. (3) C(X’VX) = C(X’). (4) ~ ( xn [~(v)]’ ) = 0. Also rank(X’VX) = p if and only if r ankX = p and C(X) n [C(V)]’ = 0. Proofs. 2. 9. Abadir and Magnus [2005: 82, 851. 10. 11. 8. 12. Abadir and Magnus [2005: 811 and Meyer [2000a: 2111. 13. Marsaglia and Styan [1974a: theorem 61. 14a.