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MATH 2107assign2_09

# MATH 2107assign2_09 - MATH 2107 LINEAR ALGEBRA II...

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MATH 2107 LINEAR ALGEBRA II ASSIGNMENT 2 DUE: December 4 at the beginning of the tutorial 1. Let = 2 2 2 2 5 4 2 4 5 A (a) Show that the matrix A is positive definite. (b) Find the Cholesky factorization of the matrix A . (c) Orthogonally diagonalize the matrix A . 2. (a) Find the matrix A of the quadratic form 3 1 2 1 2 3 2 2 2 1 8 8 11 7 9 x x x x x x x q + + + = . (b) Given that the eigenvalues of A are 3, 9 and 15. Find an orthogonal matrix P such that the change of variable Py x = that will transform the quadratic form into one with no cross product term. Give P and the new quadratic form. 3. Let = 3 4 1 6 2 1 3 8 3 6 6 1 A and ⎡− = 5 17 25 1 b (a) Find the QR-factorization of the matrix A . (b) Find the least square solution to the equation
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