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Effect of orthonormal projection on rank

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Given rank(R) and a QR factorization A = QR, what is the rank(A)


    2. Relevant equations



    3. The attempt at a solution
    I want to know if multiplication by a full rank orthonormal matrix Q and an upper trapezoidal matrix R yields rank(R)=rank(Q*R)=rank(A)

    This is mostly guesswork by me but I'd like to use it for a question I need to answer.
     
  2. jcsd
  3. Sep 14, 2009 #2

    Dick

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    Well, the rank of a matrix is the dimension of the image, right? If the image of R is a subspace S of dimension rank(R), then what's the dimension of Q(S) if Q is full rank?
     
  4. Sep 14, 2009 #3
    They are equal? As the only way Q(S) would be dissimilar would be if rank(Q)<rank(R).

    But does not the reason for this have anything to do with Q being orthnormal? Otherwise couldn't Q act on R and cause some of the image to overlap effectively reducing the rank?
     
  5. Sep 14, 2009 #4

    Dick

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    Q is full rank, so it's one to one. So yes, rank(QR)=dim(Q(S))=dim(S)=rank(R). So rank(QR)=rank(R).
     
  6. Sep 14, 2009 #5
    Thank you very much =)
     
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