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ps7 - by x> 1 Ax 2 where x = x> 1 x> 2> x 1 is an m...

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Atsushi Inoue ECG 765: Mathematics for Economists Fall 2009 Problem Set ] 7 Due in class on Tuesday, October 20 1. Examine the definiteness or semidefiniteness of the following matrix: A = 1 2 3 2 4 6 3 6 0 . (2 points) 2. Suppose that matrix A has an eigenvalue, λ . (a) Prove that one of the eigenvalues of A 2 = AA is λ 2 . (2 points) (b) Using part (a) prove that eigenvalues of an idempotent matrix is either 0 or 1. (2 points) 3. (a) Define f : < m + n → < by x > 1 Ax
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Unformatted text preview: by x > 1 Ax 2 where x = [ x > 1 x > 2 ] > , x 1 is an m × 1 vector, x 2 is an n × 1 vector, and A is an m × n matrix. Find the gradient vector and Hessian matrix of f . (2 points) (b) Let X be an m × n matrix Define g : < mn → < by a function that maps x = vec( X ) to a > Xb where a is an m × 1 vector, b is an n × 1 vector. Find the gradient vector and Hessian matrix of g . (2 points) 1...
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