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Math Help - LU-decompositions

  1. #1
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    LU-decompositions

    I have a question on a basic theorem:

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    If A is a square matrix that can be reduced to row-echelon form, without interchanging any rows then A can be factored as A=LU where U is the row-echelon form of A.
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    This seems pretty simple, but it's not entirely clear why interchanging rows is illegal. Any clarification would be appreciated.
    Last edited by adkinsjr; January 14th 2010 at 01:00 PM.
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  2. #2
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    Quote Originally Posted by adkinsjr View Post
    I have a question on a basic theorem:

    --------------------------------------------------------------------------------------------------------------------------------
    If A is a square matrix that can be reduced to row-echelon form, then without interchanging any rows then A can be factored as A=LU where U is the row-echelon form of A.
    --------------------------------------------------------------------------------------------------------------------------------

    This seems pretty simple, but it's not entirely clear why interchanging rows is illegal. Any clarification would be appreciated.
    This doesn't say anything about interchanging rows being "illegal". Nor does it say that if the row-reduction requires interchanging rows, then A cannot be factored as LU. Saying "If A then B" does NOT imply "if not A then not B". If A can be reduced to row-echelon form, then that row-echelon form is upper triangular and can be used as "U". If there were no row interchanges, the matrix you get by applying those same row operations to the identity matrix will be lower triangular and can be used as "L". If there are row interchanges, that matrix is not lower triangular and cannot be used as "L". That doesn't necessarily mean A cannot be "LU" decomposed, only that the proof would be harder.
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