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Math Help - [SOLVED] Permutation Matrices

  1. #1
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    [SOLVED] Permutation Matrices

    Hey all, this is one of my practice questions and I have no idea where to start.
    Find a 5 by 5 permutation matrix P which has 1 in its (5,4)-place and so that the smallest power of P which gives the identity matrix is P6 .
    (Hint: Consider "combining" a 2 by 2 permutation matrix with a 3 by 3 one.)


    There are several correct different solutions.


    The question has been designed so that there should be a block diagonal permutation matrix solution.



    I haven't actually encountered permutation matrices before, but I've looked up the basics (what they do, how to apply them).
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  2. #2
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    Rather than thinking about permutation matrices, think about permutations.

    We're looking for a permutation which has order 6, the hint given to us suggests using a permutation of order 2 and a permutation of order 3 together (which will after all give a permutation of order lcm(3,2) = 6)

    Now, how can we create such a permutation, given the requirement that it will shift 4->5
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  3. #3
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    Do you mean permutations as in {0,0,1} to {0,1,0} to.. etc? I'm not quite sure how you'd combine two permutations of two different orders, something like

    Perm[{0,0,1}] combined with Perm[{0,1}]?

    Which I guess would give {0,0,1,0,1} and {0,0,1,1,0}, amongst other permutations, but they're of order 5. Can you explain a little more?

    *edit* I should really add that this was designed as a 'revision problem', but we haven't been taught such a thing before (I guess it was assumed prior knowledge). So layman's terms are fine
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  4. #4
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    I ended up solving the problem by writing a brute force algorithm in Mathematica, but I'd still like to know how to do it by hand. I can post the code if anybody is interested, but it is fairly straight forward.
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  5. #5
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    Well, what is the permutation matrix for the element of S_5 which is (123)(45)
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  6. #6
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    Quote Originally Posted by SimonM View Post
    Well, what is the permutation matrix for the element of S_5 which is (123)(45)
    The matrix which changes the vector (1,2,3,4,5) -> (1,2,3,5,4) would be

    |1 0 0 0 0|
    |0 1 0 0 0|
    |0 0 1 0 0|
    |0 0 0 0 1|
    |0 0 0 1 0|

    Which has a 1 in the element (5,4). Is this what you meant?

    Or a permutation matrix which changes (123)(45) -> (45)(123) would be

    |0 0 0 0 1|
    |0 0 0 1 0|
    |1 0 0 0 0|
    |0 1 0 0 0|
    |0 0 1 0 0|
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  7. #7
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    You've got the element (45) as your first matrix, but we want (123)(45) (the matrix which permutes cyclically 1,2,3 at the same time)
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  8. #8
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    Okay,

    |0 1 0 0 0|
    |0 0 1 0 0|
    |1 0 0 0 0|
    |0 0 0 0 1|
    |0 0 0 1 0|

    Should cycle (45) as well as (123) cyclically.
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  9. #9
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    And that's my answer! Thanks for your patience.
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  10. #10
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    Excellent. You might also want to find all the other permutation matrices which work (as an extension problem)
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