MATH 2107 LINEAR ALGEBRA II ASSIGNMENT 2 DUE: December 4 at the beginning of the tutorial 1. Let ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−=222254245A(a) Show that the matrix Ais positive definite. (b) Find the Cholesky factorization of the matrix A. (c) Orthogonally diagonalize the matrix A. 2. (a) Find the matrix Aof the quadratic form 3121232221881179xxxxxxxq+−++=. (b) Given that the eigenvalues of Aare 3, 9 and 15. Find an orthogonal matrix Psuch that the change of variable Pyx=that will transform the quadratic form into one with no cross product term. Give Pand the new quadratic form. 3. Let ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−−=341621383661Aand ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=517251b(a) Find the QR-factorization of the matrix A. (b) Find the least square solution to the equation
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