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Math Help - Determining Matrix P

  1. #1
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    Determining Matrix P

    Hey there,
    So I've been given this question, determine a matrix P so that P^-1AP=B?
    The questions relate to the matrices
    A = [1 2 3]
    [0 -1 2]
    [0 0 3]

    B= [-1 -2 3]
    [0 1 -5]
    [0 0 3]

    The eigenvalues for the matrices are the same: lamda equals 3, 1 and -1.
    And the eigenvectors for lamda equals 1 and -1 are the same for both matrices: for 1: 1, 0, 0 and for -1: -1, 1, 0.
    The eigenvectors for lamda equals 3 in A: 2, 1/2, 1.
    The eigenvectors for lamda equals 3 in B: 2, -5/2, 1.

    So I've done a lot of working out so far but now I'm stuck! Can anyone let me know how to go about finding matrix P.
    I thought I was just meant to put the eigenvectors of A into a matrix:
    e.g. [1 -1 2]
    [0 1 1/2]
    [0 0 1]
    However, that didn't work so I'm really unsure now!
    Thanks heaps in advance.
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  2. #2
    MHF Contributor
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    Re: Determining Matrix P

    Hey grooverandshaker.

    If you have the eigen-vectors then P is just made up by using the eigenvectors corresponding to a particular eigen-value while D has the eigenvalues in the diagonal (with zeroes everywhere else).

    Try putting the eigenvectors on the appropriate rows of P, calculate P inverse and then do the multiplication and see if you get B back.

    If you ever want to double check your work use something like Octave which does a lot of MATLAB does only its free and open source. (GUIOctave is the GUI front end for it and is a separate download).
    Thanks from topsquark
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  3. #3
    MHF Contributor

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    Re: Determining Matrix P

    You don't need to calculate eigenvectors and eigenvalues. Note that B can be derived from A by "row operations".

    1) Multiply row 1 by -1.
    2) Multiply row 2 by -1.
    3) Add -1 times row 3 to row 2.

    Each of those corresponds to an "elementary matrix", the matrix you get by applying the same row operation to the identity matrix, and their product gives "P".
    Thanks from topsquark
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  4. #4
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    Re: Determining Matrix P

    Thanks so much for your help. I don't really understand what the row operation for row 3 is though? Can anyone explain?
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