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Math Help - permutation matrix question

  1. #1
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    permutation matrix question

    the question highlighted below is in two parts, really have no clue how to start it, any help would be greatly appreciated!!

    a. Recall that an elementary permutation matrix is an n x n matrix which is In except that two rows of
    In have been swapped. If P_{i;j} is the elementary permutation matrix where rows i and j have been swapped
    and A is a matrix (with n rows), describe the relationship between A and P_{i;j}A.

    b. Let A be the matrix:

    A=\left[\begin{array}{ccc}1&2&3\\2&4&6\\1&3&5\end{array}\r  ight]

    Find a permutation matrix P, a lower triangular matrix L and an upper triangular matrix U so that
    A = P^{T}LU:

    thank you!
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  2. #2
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    Quote Originally Posted by situation View Post
    the question highlighted below is in two parts, really have no clue how to start it, any help would be greatly appreciated!!

    a. Recall that an elementary permutation matrix is an n x n matrix which is In except that two rows of
    In have been swapped. If P_{i;j} is the elementary permutation matrix where rows i and j have been swapped
    and A is a matrix (with n rows), describe the relationship between A and P_{i;j}A.
    Have you looked at an example? Write down any permutation matrix, P, any matrix A, and multiply them. Compare A and PA. The answer should be obvious.

    b. Let A be the matrix:

    A=\left[\begin{array}{ccc}1&2&3\\2&4&6\\1&3&5\end{array}\r  ight]

    Find a permutation matrix P, a lower triangular matrix L and an upper triangular matrix U so that
    A = P^{T}LU:

    thank you!
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Have you looked at an example? Write down any permutation matrix, P, any matrix A, and multiply them. Compare A and PA. The answer should be obvious.
    ok so am i right in saying that if i took any identity matrix, swapped any of the two rows, that would be denoted a permutation matrix?

    eg. \left(\begin{array}{ccc}1&0&0\\0&0&1\\0&1&0\end{ar  ray}\right) would be a permutation matrix on the identity matrix for a 3x3??
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  4. #4
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    You're correct. However, your problem has asked you to compare the result when you take the matrix you exhibited in your last post and left-multiply it with A, versus just plain A itself. That is, if P is the matrix in your last post, then what's the difference between A and PA?
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