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Math Help - eigenvector problem

  1. #1
    Newbie
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    Jan 2009
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    eigenvector problem

    Hi, I was wondering if you guys would be able to help me out with a problem I've been struggling with for a few days.

    Assuming M = 2x2 matrix with a single eigenvalue c, with eigenvector x, and that y is a vector not equal to 0 which is not an eigenvector. How can I show that x and y are linearly independent?
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
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    France
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    1,458
    Hi

    M being a matrix with a single eigenvalue c, with eigenvector x you have
    Mx = cx

    Suppose x and y are not linearly independant. There exists a real k such as y=kx
    My = M(kx) = k(Mx) = k(cx) = c(kx) = cy
    Then y would be an eigenvector of M which is not the case
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