Fundamental equations

You just learned that if A stretches and sends an orthonormal basis {u_1, u_2} to an orthonormal basis {v_1, v_2} then A has an SVD of A = (v    v ) (s      ) (u )       1    2    1   0     1                      s    u                0     2    2.

If on the other hand you start with orthonormal bases {u_1, u_2} and {v_1, v_2} and define A = (v    v ) (s      ) (u )       1    2    1   0     1                      s    u                0     2    2 then you'll know that A will stretch and send the orthonormal basis {u_1, u_2} to the orthonormal basis {v_1, v_2}.  

In particular you'll know A u_1 = s_1v_1 and A u_2 = s_2v_2.  These are the FUNDAMENTAL EQUATIONS for an SVD.

Define A = ( 3  4  ) (3   0) ( 5   12  ) .               -  -         ...    -- -                              --- --                5 5                               13 13
You can check that both {(3 ) , (4)}   -      -   5      5    -4     3   --     -   5      5 and {(5 ), (12)}   --    --   13    13    12    5   --    --   13    13 are orthonormal bases for ^2.  This tells you that you've got an SVD for A and that the fundamental equations hold.  So, A (5 ) = 3 (3 ) = (9  )    --       -      -    13       5      5     12       -4     -12    --       --     ---    13       5       5.

(Q20)  And A (-12) =    ---    13     5    --    13


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Copyright 2007 Todd Will
Created by Mathematica  (April 14, 2007)