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Need Help with positive definite matrices

  1. Aug 8, 2013 #1
    1. The problem statement, all variables and given/known data
    If A is positive definite, show that ## A = C C^T ## where ## C ## has orthogonal columns.

    3. The attempt at a solution

    So, I've got the first part figured out. Because ## A ## is symmetric, an orthogonal matrix ## P ## exists such that ## P^TA P = D = diag(\lambda_1,...,\lambda_n) ## where ## \lambda_i > 0 ## because A is positive definite. Next I've defined ## B = diag(\sqrt{\lambda_1},...,\sqrt{\lambda_n}) ## Then I wrote ## C = P^TB P ## then ## C C^T = (P^T B P) (P^T B P)^T = (P^T B P) P^T B^T P = P^T B B P = P^T D P = A ##

    So I'm at the last step and I'm stuck on how to show that C has orthogonal columns. Any hints?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 8, 2013 #2

    Office_Shredder

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    C has orthogonal columns if and only if CtC is a diagonal matrix (you should be able to check that this claim is true easily- the off diagonal entries of Ct C are the inner products of the columns of C).

    Also PDPt = A, not PtDP.
     
  4. Aug 9, 2013 #3
    Right, that's just sloppyness on my part typing that out. I haven't learned anything about inner products yet. Is there a way to look at this differently?
     
  5. Aug 9, 2013 #4

    Office_Shredder

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    Inner product is the same as dot product. The typical definition of orthogonal vectors is that their dot product is zero, if you're working with a different definition you'll have to say what it is
     
  6. Aug 9, 2013 #5
    I just read your first reply again. I get it now. I was pretty tired last night. Thanks a lot!
     
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