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Math Help - basis for the eigenspace

  1. #1
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    basis for the eigenspace

    | -3 , 0 , 0, 0 |
    | 0 , 2, 1 , 0| = A
    | 0 , 0, 2, 0 |
    | 0 , 0 , 0 , 2|

    The characteristic polynomial of A is (t + 3)(t − 2)^3. Find a basis for the eigenspace of A corresponding to the
    eigenvalue λ = 2.

    The answer is

    {[0,1,0,0]^T, [0,0,0,1]^T}

    Any help would be appreciated.
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  2. #2
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    Re: basis for the eigenspace

    Have you tried just using the DEFINITIONS?

    An eigenvector, v, of A with eigenvalue 2 must satisfy Ax= 2x so
    \begin{bmatrix} 3 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 2 & 0  \\ 0 & 0 & 0 & 2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \\ u\end{bmatrix}= \begin{bmatrix} 2x \\ 2y \\ 2z \\ 2u\end{bmatrix}
    which gives the equations -3x= 2x, 2y+ z= 2y, 2z= 2z, 2u= 2u. What do you get when you try to solve those equations?
    Last edited by HallsofIvy; June 5th 2012 at 05:04 PM.
    Thanks from killingphill
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  3. #3
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    Re: basis for the eigenspace

    I am, however, concerned about your choice of username: "killingphil". Whose is this "phil" and why are you so angry at him?
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