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Orthogonal projectors (minimization and variational problem)

  1. Sep 10, 2009 #1
    1. The problem statement, all variables and given/known data
    S1 is in subspace of C^n. P unique orthogonal projector P : C^n -> S1, and x is in range of C^n. Show that the

    minimization problem: y in range of S1 so that:
    2norm(x-y) = min 2norm(x-z)
    where z in range of S1

    and

    variational problem: y in range of S1 so that:
    (x-y)*z = 0 (* is the hermitian so this is an inner product)
    for each z in range of S1

    have the same unique solution y = Px


    2. Relevant equations
    P^2-P = 0
    Range(P) = S1 ?


    3. The attempt at a solution
    I have no idea what's going on here. I am using Numerical Linear Algebra by Trefethen and Bau but there's nothing about minimization or variational problems in the book =/ any hint to get me started would be much appreciated.

    For the variational problem this might be one way but it doesn't show that it's unique and I'm not even sure it's correct:
    (x-y)*z = 0
    Set y = Px and multiply by P
    (Px-P^2x)*z = 0
    but P^2 = P so (Px-P^2x) = 0 and therefore LHS = RHS

    Clueless on the first one.
     
  2. jcsd
  3. Sep 10, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    I think the fact you might be missing is that x-Px is orthogonal to any element y of S1. So (x-Px)*y=0 for all y in S1. That's what it means to be an ORTHOGONAL projector. For the first part, write norm(x-z)=norm((x-Px)+(Px-z)). Can you see the Pythagorean theorem in there?
     
  4. Sep 10, 2009 #3
    Wow, I didn't know maths could be fun :P thanks a lot.
     
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