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Math Help - Linear Algebra Eigenvectors -- question on x1+x2+x3 = 0 and more

  1. #1
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    Question Linear Algebra Eigenvectors -- question on x1+x2+x3 = 0 and more

    Hello.

    Okay, I'm struggling with eigenvectors -- I have already searched the forum and found a thread here (http://www.mathhelpforum.com/math-he...n-problem.html) which is more or less about the subject I'm interested in. But I still need to know how the answer was found.


    Let A = [1 1 1]
    ----- = [0 0 0]
    ----- = [0 0 0]

    This is a RREF matrix, which I want to 'extract' an eigenvector from...

    TD! posted that the eigenvectors are e.g. (1,0,-1) and (1,-1,0). What I want to know is how they are found?


    Also - I would like to know more about eigenvectors of this matrix...

    B = [0 1 0]
    - = [0 0 1]
    - = [0 0 0]

    I see x2 = x3 = 0, so I guess the eigenvector is [? 0 0], but what is x1? I think x1 is supposed to be x1 = 1, but why? I don't get why..?



    Thanks in advance!
    Simon DK.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Taguig City, Philippines
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    Quote Originally Posted by sh01by View Post
    Hello.

    Okay, I'm struggling with eigenvectors -- I have already searched the forum and found a thread here (http://www.mathhelpforum.com/math-he...n-problem.html) which is more or less about the subject I'm interested in. But I still need to know how the answer was found.


    Let A = [1 1 1]
    ----- = [0 0 0]
    ----- = [0 0 0]

    This is a RREF matrix, which I want to 'extract' an eigenvector from...

    TD! posted that the eigenvectors are e.g. (1,0,-1) and (1,-1,0). What I want to know is how they are found?
    it can be found by solving this equation:
    det (xI - A) = 0, then solve for x..


    Quote Originally Posted by sh01by View Post
    Also - I would like to know more about eigenvectors of this matrix...

    B = [0 1 0]
    - = [0 0 1]
    - = [0 0 0]

    I see x2 = x3 = 0, so I guess the eigenvector is [? 0 0], but what is x1? I think x1 is supposed to be x1 = 1, but why? I don't get why..?



    Thanks in advance!
    Simon DK.
    actually, it is one of the eigenvectors. the eigenvectors of the matrix is in the set {[r 0 0] | r is in the Field}, this set is called the eigenspace of the eigenvalue of the matrix.
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  3. #3
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    Hmm...

    "it can be found by solving this equation:
    det (xI - A) = 0, then solve for x.."

    Isn't that the characteristic polynomial to find the eigenvalues?
    If it is, then that is not really what I'm searching for... If it isn't, then I just don't understand entirely, maybe you could do an example?
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by sh01by View Post
    "it can be found by solving this equation:
    det (xI - A) = 0, then solve for x.."

    Isn't that the characteristic polynomial to find the eigenvalues?
    If it is, then that is not really what I'm searching for... If it isn't, then I just don't understand entirely, maybe you could do an example?
    okay, after finding the eigenvalues, the eigenvectors are found by looking for the eigenspace, that is, solving for det (\lambda I - A)X = 0
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