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Math Help - Eigenvalues/Eigenvectors

  1. #1
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    Eigenvalues/Eigenvectors

    How do I determine what the original matrix was that yielded these two eigenvalues with the corresponding eigenvectors:

    \lambda_1 = -3 Eigenvector: [0,1]

    \lambda_2 = 2 Eigenvector: [1,0]
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  2. #2
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    Quote Originally Posted by Ideasman View Post
    How do I determine what the original matrix was that yielded these two eigenvalues with the corresponding eigenvectors:

    \lambda_1 = -3 Eigenvector: [0,1]

    \lambda_2 = 2 Eigenvector: [1,0]
    Work:

    (\lambda + 3)(\lambda - 2) = \lambda^2 + \lambda - 6

    (a - \lambda)(d - \lambda) - bc = \lambda^2 + \lambda - 6

    \lambda^2 - a\lambda - d\lambda + ad - bc = \lambda^2 + \lambda - 6

    I thought the following matrix would work, but it didn't :

    \left[ \begin {array}{cc} -2&2\\\noalign{\medskip}2&1\end {array}<br />
 \right]

    I got the right eigenvalues, but not the right eigenvectors, grr.
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  3. #3
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    See if this helps:

    det\begin{bmatrix}{\lambda}&1\\{\lambda}+6&{\lambd  a}+2\end{bmatrix}={\lambda}^{2}+{\lambda}-6

    The correct matrix will work if Ax={\lambda}x

    Where x is the eigenvector and \lambda is the eigenvalue.

    You are given eigenvalues and eigenvectors.
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  4. #4
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    Quote Originally Posted by galactus View Post
    See if this helps:

    det\begin{bmatrix}{\lambda}&1\\{\lambda}+6&{\lambd  a}+2\end{bmatrix}={\lambda}^{2}+{\lambda}-6

    The correct matrix will work if Ax={\lambda}x

    Where x is the eigenvector and \lambda is the eigenvalue.

    You are given eigenvalues and eigenvectors.
    I tried. I can't figure it out.

    EDIT: Yay I figured it out. Good old PDP^(-1). The matrix, incase you were wondering, is [[2,0],[0,-3]]
    Last edited by Ideasman; December 11th 2007 at 07:38 PM.
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