Proof of Singular Value Decomposition

  • Thread starter the_kid
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Homework Statement


A is arbitrary linear map of the complex vector space s.t. A: V-->V.

1) Show that there exist unitary matrices s.t A=V*DU where D is diagonal and its entires are non-negative.
2) Show that part 1 holds if and only if there exist orthonormal bases {u_i} and {v_i} s.t. Au_i=d_i v_i and A*v_i=d_i u_i where u_i is the i-th column of U* and v_i is the i-th column of V*.


Homework Equations





The Attempt at a Solution


I was able to successful prove part 1. However, I am really stuck on part 2. I know I need to show both directions since it is an iff proof. So, if I assume these bases exist, I need to be able to derive the SVD.
 

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