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Math Help - Generalized Eigenvectors from repeated Eigenvalues

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    Generalized Eigenvectors from repeated Eigenvalues

    Could anyone explain to me how you get the eigenvectors of the following matrix

    1 2 3
    0 1 4
    0 0 1

    I know this is a triangular matrix and the eigenvalues are 1, 1, 1 but when I check my work on matlab, it gives me the eigenvectors as [1 0 0] [-1 0 0] [1 0 0] and these are not what I calculate. I first try to find the exact eigenvector and get [a 0 0] then I set the (A-(lamdai)I) equation equal to the exact eigenvector I got to find the first generalized eigenvector. What am I doing wrong? Thanks!
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  2. #2
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    Quote Originally Posted by rustybladz View Post
    Could anyone explain to me how you get the eigenvectors of the following matrix

    1 2 3
    0 1 4
    0 0 1

    I know this is a triangular matrix and the eigenvalues are 1, 1, 1 but when I check my work on matlab, it gives me the eigenvectors as [1 0 0] [-1 0 0] [1 0 0] and these are not what I calculate. I first try to find the exact eigenvector and get [a 0 0] then I set the (A-(lamdai)I) equation equal to the exact eigenvector I got to find the first generalized eigenvector. What am I doing wrong? Thanks!
    It looks as though it's matlab that's wrong, not you. There is a triple eigenvalue \lambda=1, but only a one-dimensional eigenspace, namely multiples of [1 0 0]. Matlab is giving you this vector three times over.

    You are following the correct procedure for finding the generalised eigenvectors.
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