Hand computation of SVD

(Q46) Every matrix A 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 A = ( 54  -28 )                -25 0                -6  42 by hand.  An SVD for A will have the form A = (v    v    v ) (s      ) (u )       1    2    3    1   0     1                           s    u                     0     2    2                       0    0.
a.  Start with the vectors p_1 = (t    )          1 - tand p_2 = (t - 1)          t.  For any value of t these vectors are orthogonal.  Find a value of t so that A p_1 and A p_2 are orthogonal.  I.e., compute A p_1 and A p_2, find their dot product, and then find a value of t that makes the dot product zero.
b.  Once you've found t, you can get u_1 and u_2 by normalizing (making unit vectors out of) p_1 and p_2.
c.  You can get v_1 and v_2 by normalizing (making unit vectors out of) (Ap) _1 and (Ap) _2.
d.  You can get s_1by computing how much A stretched p_1.  I.e. compute s_1 = A p_1/p_1, similarly for s_2.
e. Use the cross product to get a vector v_3 to complete the orthonormal basis of ^3.


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Copyright 2007 Todd Will
Created by Mathematica  (April 15, 2007)