# Proof of Singular Value Decomposition

1. Sep 26, 2012

### the_kid

1. The problem statement, all variables and given/known data
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*.

2. Relevant equations

3. 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.