By Bapat R.P.

The most objective of Linear Algebra and Linear versions is to supply a rigorous creation to the elemental facets of the idea of linear estimation and speculation trying out. the required must haves in matrices, multivariate general distribution and distributions of quadratic varieties are built alongside the way in which. The ebook is aimed toward complicated undergraduate and first-year graduate masters scholars taking classes in linear algebra, linear types, multivariate research, and layout of experiments. it may even be of use to analyze mathematicians and statisticians as a resource of normal effects and difficulties.

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Iii) Let βi be the BLUE of βi and let β be the column vector with components β1 , . . , βp . Then β (X X)−1 X y. The dispersion matrix of β is σ 2 (X X)−1 . (iv) The BLUE of β is β with variance σ 2 (X X)−1 . Parts (iii) and (iv) constitute the Gauss–Markov theorem. Problems 1. Consider the model E(y1 ) 2β1 − β2 − β3 , E(y2 ) β2 − β4 , E(y3 ) β2 + β3 − 2β4 with the usual assumptions. Describe the estimable functions. 38 2. Linear Estimation 2. Consider the model E(y1 ) β1 + β2 , E(y2 ) β1 − β2 , E(y3 ) β1 + 2β2 with the usual assumptions.

Un ), bij and hence C is positive semidefinite. The general case is obtained using the spectral decomposition of B. 33. Hint: For t > 0, the n × n matrix (t xi +xj ) is positive semidefinite. Now use 1 xi +xj −1 1 dt. the fact that xi +x 0 t j 34. Using the spectral theorem we may assume, without loss of generality, that X diag(x1 , . . , xn ). Let XY + YX Z. Then yij (xi +xj ) zij , and hence zij for all i, j . Now use the preceding two exercises. yij xi +xj 31. Assume, without loss of generality, that A 1 1 1 1 (A 2 − B 2 ).

Similarly, a symmetric matrix is positive semidefinite if and only if its eigenvalues are all nonnegative. 9. If A is positive semidefinite, then there exists a unique positive semidefinite A. The matrix B is called the square root of A and is matrix B such that B2 denoted by A1/2 . Proof. There exists an orthogonal matrix P such that (6) holds. Since A is positive semidefinite, λi ≥ 0, i 1, . . , n. Set 1 B 1 Pdiag(λ12 , . . , λn2 )P . Then B2 A. To prove the uniqueness we must show that if B, C are positive semidefinite matrices satisfying A B2 C2 , then B C.