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Oblique projector

  1. Sep 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the oblique projector P so where range(P) = range(U) and range(I-P) = range(W)


    2. Relevant equations
    P^2-P = 0
    range(I-P) = null(P)


    3. The attempt at a solution
    It seems that U and W are complementary subspaces. According to:
    http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.4500v4.pdf
    U*(V*U)t*V
    where t is the Moore-Penrose inverse. The task is to be done in Matlab. My problem is that V and U cannot be multiplied due to their sizes. The sizes are for example (5,3) and (5,8) with the rows in common. I tried adding on the identity to fill up space in U and also tried to add null space but this did not work. I always seem to end up with a projector P so that:
    P^2-P = 0
    PA = A
    but PB != 0
    and (I-P)B != B

    Any ideas?
     
  2. jcsd
  3. Sep 10, 2009 #2
    p = [U 0] [U V]^-1

    According to "Generalized inverses: theory and applications" by AvAdi Ben-Israel, Thomas Nall Eden Greville

    0 is nullspace

    And sure as hell it works =D
     
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