Hand computation of SVD

(Q46) Every matrix has a singular value decomposition and to find it you usually have to use a machine. But if you like, this problem will show you how to find an SVD for by hand. An SVD for will have the form .

a. Start with the vectors and . For any value of these vectors are orthogonal. Find a value of so that and are orthogonal. I.e., compute and , find their dot product, and then find a value of that makes the dot product zero.

b. Once you've found , you can get and by normalizing (making unit vectors out of) and .

c. You can get and by normalizing (making unit vectors out of) and .

d. You can get by computing how much stretched . I.e. compute , similarly for .

e. Use the cross product to get a vector to complete the orthonormal basis of .

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Copyright © 2007 Todd Will

Created by
Mathematica (April 15, 2007)