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)