1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Permutation Matrices
  1. Permutation matrices (Replies: 1)

  2. Permutable matrices (Replies: 3)

  3. The Matrices (Replies: 12)

  4. Permutations ? (Replies: 3)

Loading...