P3 [Discovery proof of SVD for a 2 2 matrix] Let A a) Verify that the...

 

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P3 [Discovery proof of SVD for a 2 2 matrix] Let A a) Verify that the vectors vi [cos 0 sin 0 and v2 = [-] sin 0 cos 0 are orthonormal for any choice of the parameter 0 <0 < 2. b) (MATLAB) Plot the angle between Av and Av as a function of 0. For how many angles are Avi and Av2 orthogonal? c) Show that Av, and Av are orthogonal if and only if tan 20 = 3. How many angles exist in the interval [0, 2) satisfying this condition? Explain why this corresponds to choosing {v1, v2}. d) (MATLAB) For all the angles obtained in c) numerically determine ||Av ||2 and ||Av2||2. e) Assume that tan 20 = 3. Show that = ||Avi ||2||Av2||2 = 02 provided cos 20 > 0. f) Assume tan 20 = with cos 20 0 and let u Av2. Explain why = = Avi, uz U = = [u | u], = | 0 002] and V = [p| v2] provide a SVD decomposition A = UV of A. g) (MATLAB) Numerically compare the decomposition obtained in f) with the results of the MATLAB function svd for the two angles 0 satisfying tan 20 = with cos 20 0. P3 [Discovery proof of SVD for a 2 2 matrix] Let A a) Verify that the vectors vi [cos 0 sin 0 and v2 = [-] sin 0 cos 0 are orthonormal for any choice of the parameter 0 <0 < 2. b) (MATLAB) Plot the angle between Av and Av as a function of 0. For how many angles are Avi and Av2 orthogonal? c) Show that Av, and Av are orthogonal if and only if tan 20 = 3. How many angles exist in the interval [0, 2) satisfying this condition? Explain why this corresponds to choosing {v1, v2}. d) (MATLAB) For all the angles obtained in c) numerically determine ||Av ||2 and ||Av2||2. e) Assume that tan 20 = 3. Show that = ||Avi ||2||Av2||2 = 02 provided cos 20 > 0. f) Assume tan 20 = with cos 20 0 and let u Av2. Explain why = = Avi, uz U = = [u | u], = | 0 002] and V = [p| v2] provide a SVD decomposition A = UV of A. g) (MATLAB) Numerically compare the decomposition obtained in f) with the results of the MATLAB function svd for the two angles 0 satisfying tan 20 = with cos 20 0.