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Homework Help: Permutation Matrices

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Supposing P is a permutation matrix, I have to show that PT(I+P) = (I+P)T. Is there any general form of a permutation matrix I should use here as permutation matrices of a dimension can come in various forms.

    2. Relevant equations



    3. The attempt at a solution

    I did this letting P = [0, 1| and it did indeed work out fine.
    |1, 0]
     
  2. jcsd
  3. Apr 3, 2013 #2
    Doesn't P have to be a square matrix? Or maybe I'm just not familiar with your notation?
     
  4. Apr 3, 2013 #3
    Yes, it does have to be square. I know it must also have a single 1 in each row and column, the rest zeros. But this can happen in multiple ways correct? So is there not a generalized form for a permutation matrix?

    ie. [1 0|
    |0 1]

    OR

    [0 1|
    |1 0]
     
  5. Apr 3, 2013 #4
    What is the definition of a "permutation matrix"?
     
  6. Apr 3, 2013 #5
    It is a matrix created from the identity by arranging rows and columns. It has a single 1 in each row and column; the rest are zeros.
     
  7. Apr 3, 2013 #6
    OK. Then that's the only thing you can use. You can't pick a special form of ##P## and prove it for that. You need to prove it for all possible ##P##.

    That said, are you familiar with elementary row and column transformations? This can help you. Why? Because any permutation matrix can be made from the identity matrix by just exchanging a few columns and rows.
     
  8. Apr 3, 2013 #7
    I know that to get P I can multiply I by an elementary matrix. I understand the concept, but am unsure how I am supposed to go about proving this question. The transpose of the permutation will always just be the permutation.. correct?
     
  9. Apr 3, 2013 #8
    The idea is to prove the equation ##P^T(I+P) = (I+P)^T## first for elementary matrices that switch a row or a column. Then you should only show that if two matrices ##P## and ##Q## satisfy the equation, then so does their product. I claim that this shows that the equation holds for each permutation matrix.
     
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