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Calculation and Uniqueness of Smith Normal Forms

  1. Jan 13, 2012 #1

    jj4

    User Avatar

    FYI this is a homework problem which I already have the answer to but would like to clarify some points on.

    1. The problem statement, all variables and given/known data

    Find the Smith Normal Form of the matrix

    [tex]
    \left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right]
    [/tex]

    over the ring of integers.

    2. Relevant equations

    As I understand it Smith Normal Forms are unique up to multiplication by a unit in R.

    3. The attempt at a solution

    I have performed a series of elementary row and column operations to get to the following Smith Normal Form:

    [tex]
    \left[ \begin{array}{cccc} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right]
    [/tex]

    However the "answer" given is actually:

    [tex]
    \left[ \begin{array}{cccc} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right]
    [/tex]

    The only units in the ring are 1 and -1 so the answers are not equivalent and presumably I have gone wrong somewhere. Can someone help me out here. My problem is not only related to this example, I have the same issue with others. Is it because my Smith Normal Form and the original matrix are not actually equivalent?
     
  2. jcsd
  3. Jan 14, 2012 #2

    jj4

    User Avatar

    I'm just looking for a basic explanation of how to calculate Smith Normal Forms. Is this thread in the correct sub-forum?
     
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