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Math Help - Help with Eigenvectors

  1. #1
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    Help with Eigenvectors

    Hello,

    Was wondering if anyone could help, i am trying to work out eigenvectors after working out the 3 eigenvalues for the below matrix:
    [5 3 2
    1 4 6
    9 7 3]
    My eigenvectors are correct however for 2 of the eigenvalues, my eigenvectors have opposite signs.
    I have checked these in MathCAD so i know that the eigenvectors are correct however for some reason some of the signs are opposite.
    I have attached my workings for the eigenvalue 1.65, the eigenvectors shown have opposite signs, can anyone spot why.

    Thanks so much for your help.


    Attachment 22720Attachment 22719
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  2. #2
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    Re: Help with Eigenvectors

    Quote Originally Posted by hfullwood View Post
    Hello,

    Was wondering if anyone could help, i am trying to work out eigenvectors after working out the 3 eigenvalues for the below matrix:
    [5 3 2
    1 4 6
    9 7 3]
    My eigenvectors are correct however for 2 of the eigenvalues, my eigenvectors have opposite signs.
    I have checked these in MathCAD so i know that the eigenvectors are correct however for some reason some of the signs are opposite.
    I have attached my workings for the eigenvalue 1.65, the eigenvectors shown have opposite signs, can anyone spot why.

    Thanks so much for your help.


    Attachment 22720Attachment 22719
    If x is an eigenvector then so is x. Both answers are equally correct.
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  3. #3
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    Re: Help with Eigenvectors

    Thanks so much for your reply.
    Do you know if ive worked out the eigenvectors for all three eigenvalues and only two of sets have opposite signs, if ive followed the same method, would the other not come out with opposite signs too?
    Sorry if this is an obvious question, why does the sign of the eigenvector not matter?

    Thanks again...Hayley
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  4. #4
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    Re: Help with Eigenvectors

    suppose v is an eigenvector for the matrix A.

    then Av = λv, for some eigenvalue λ.

    but A(cv) = c(Av) = c(λv) = λ(cv), for any scalar c ≠ 0, so cv is likewise an eigenvector for A.

    in other words, Au = λu for all u in span({v}).

    in particular, all of the above is true when c = -1, as -1 is a non-zero scalar (in any field where char(F) ≠ 2).
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