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Math Help - permutation matrix

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    permutation matrix

    Prove that permutation matrices can be written as the product of transpositions.

    So geometrically, we can represent transpositions as lines between two "poles" or sticks. And completing the cycle defines a permutation?
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    Quote Originally Posted by manjohn12 View Post
    Prove that permutation matrices can be written as the product of transpositions.

    So geometrically, we can represent transpositions as lines between two "poles" or sticks. And completing the cycle defines a permutation?
    for any \alpha, \beta \in S_n, let P_{\alpha}, P_{\beta} be the corresponding permutation matrices. then P_{\alpha}P_{\beta}=P_{\beta \alpha }. now use this fact that every \sigma \in S_n is a product of transpositions.
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    because every transposition matrix is a permutation?
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    Quote Originally Posted by manjohn12 View Post

    because every transposition matrix is a permutation?
    ok, how do you basically define a transposition matrix?
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    Quote Originally Posted by NonCommAlg View Post
    for any \alpha, \beta \in S_n, let P_{\alpha}, P_{\beta} be the corresponding permutation matrices. then P_{\alpha}P_{\beta}=P_{\beta \alpha }. now use this fact that every \sigma \in S_n is a product of transpositions.

    Is there not a neat way of showing the result for permutation matrices without going back into S_n?
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    Quote Originally Posted by NonCommAlg View Post
    ok, how do you basically define a transposition matrix?
    by interchanging two rows of the identity matrix
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    Quote Originally Posted by manjohn12 View Post

    by interchanging two rows of the identity matrix
    right! and when you interchange the rows i,j, you exactly get the permutation matrix P_{\sigma}, where \sigma=(i \ j). so, every transposition matrix is a permutation.
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