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Math Help - finding eigenvectors

  1. #1
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    finding eigenvectors

    For the given matrix


    2 0 -2
    0 -1 0
    -2 0 -1


    I want to find eigenvectors with a length of 1.

    So the eigenvalues are -1,-2,3

    solving for each one i got x=0, z=0 for -1
    x,y,z = 0 for -2
    and x = -2z, y=0 for 3

    so i guess the only eigenvector with length 1 would be the when the norm of the last solution equals to 1?
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  2. #2
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    Re: finding eigenvectors

    Quote Originally Posted by Kuma View Post
    For the given matrix


    2 0 -2
    0 -1 0
    -2 0 -1


    I want to find eigenvectors with a length of 1.

    So the eigenvalues are -1,-2,3

    solving for each one i got x=0, z=0 for -1
    x,y,z = 0 for -2
    and x = -2z, y=0 for 3

    so i guess the only eigenvector with length 1 would be the when the norm of the last solution equals to 1?
    Whatever eigenvector has a norm of 1. There is only 1 the other two of a norm of \sqrt{5}
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  3. #3
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    Re: finding eigenvectors

    if v is an eigenvector, so is av, for any non-zero a.

    so if a given eigenvector doesn't have norm (length) 1, just consider v/|v|. since v is an eigenvector, v is non-zero, so |v| is non-zero.
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  4. #4
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    Re: finding eigenvectors

    Quote Originally Posted by Kuma View Post
    For the given matrix


    2 0 -2
    0 -1 0
    -2 0 -1


    I want to find eigenvectors with a length of 1.

    So the eigenvalues are -1,-2,3

    solving for each one i got x=0, z=0 for -1
    x,y,z = 0 for -2
    Nonsense! The definition of "eigenvalue" is that there exist a non-zero vector such that Av= lambda v.

    and x = -2z, y=0 for 3

    so i guess the only eigenvector with length 1 would be the when the norm of the last solution equals to 1?
    You can find a non-zero eigenvector for each eigenvalue. Divide each by its norm to get length 1.
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