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Math Help - how to determine eigenvectors of 2x2 matrices like MatLab does?

  1. #1
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    how to determine eigenvectors of 2x2 matrices like MatLab does?

    Hi I am attempting to determine the eigenvalues and eigenvectors of a 2x2 matrix.

    So how to do the following Matlab instruction "manually":

    >> A = [1 1; 1 2]

    A =

    1 1
    1 2

    >> [V, D] = eig(A)

    V =

    -0.8507 0.5257
    0.5257 0.8507


    D =

    0.3820 0
    0 2.6180

    >>

    I already managed to determine the diagonal matrix D with the eigenvalues but
    how can I get the matrix v that contains the eigenvectors?
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  2. #2
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    solve for X for each eig. value: (A-(eig. val.)I)X=0
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  3. #3
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    so by inserting 0.38 into the equation I get:
    0.62x1 + x2 = 0
    x1 + 1.62x2 = 0
    sorry for the stupid question but
    while trying to solve this I cannot get one of the already determined eigenvectors using MatLab?
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  4. #4
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    try to work with the exact eigenvalues (3+-sqrt(5))/2
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  5. #5
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    no the problem is:
    it's all zero
    0.62x1 + x2 = 0 (I)
    x1 + 1.62x2 = 0 (II)

    by extending (I) with 0.62^-1 I get
    x1 + 1.62x2 = 0
    so (II) - (I) is zero at all !!! so how to
    get

    -0.8507 0.5257
    0.5257 0.8507 ??
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  6. #6
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    Of course both equations are linear dependent (because the determinant is zero after subracting the eigenvalues).

    Just solve

    0.62x1 + x2 = 0
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  7. #7
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    hmmh sorry but how to get such a vector out of:

    0.62x1 + x2 = 0 it's an equation with 2 unkown components x1 and x2
    where is the trick I don't take into account until now?
    So what brings me from 0.62x1 + x2 = 0
    to V = -0.8507 0.5257 0.5257 0.8508 ?
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  8. #8
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    let us say x1=k, then x2=-0.62k
    eigenvevtor=[k -0.62k] which is equivalent to [-0.8507 0.5257]
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  9. #9
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    ok i already realized there is a connection between these values
    but why did MatLab chose -0.8507 0.5257 it seems to be normalized to 1 sqrt(x1^2 +x2^2)?
    why can't I just say hej k= is -1 and so
    the vector would be [-1 0.62]????
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  10. #10
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    every multiple of [-0.8507 0.5257] is an eigenvector.
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  11. #11
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    so why does MatLab indicate this vector as my eigenvector and no multiple of it?
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  12. #12
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    Matlab has to make a choice, don't know why they choose this vector.
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  13. #13
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    Re: how to determine eigenvectors of 2x2 matrices like MatLab does?

    thx I think that's enough I have to know (for now) !
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