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Homework Help: Eigen Values of a 2x2 Matrix

  1. Aug 4, 2010 #1
    1. The problem statement, all variables and given/known data

    A=[1 0] Calculate
    [2 3]
    a) Eigenvalues of A
    b) Eigenvectors of A
    c) Eigenvalues and eigenvectors of A^3

    3. The attempt at a solution
    I had no idea what I was doing, but I saw someone attempt one somewhere and used the same method

    Getting x=3 and 1 for part a)

    However, I have no idea if this is correct, or even if it is in the correct format. Any help would be greatly appreciated.
     
  2. jcsd
  3. Aug 4, 2010 #2
    The eigenvalues of a matrix can be found as follows:

    [tex] A\vec{x} = \lambda\vec{x} [/tex]

    [tex] (A - \lambda I) \vec{x} = 0 [/tex]

    Now we know that this equation will only have a nontrivial solution if:

    [tex] det(A - \lambda I) = 0 [/tex]

    So to look at your question, we consider:

    [tex] \left|\begin{array}{cc}1-\lambda&0\\2&{3-\lambda} \end{array}\right| = 0 [/tex]

    [tex] (1 - \lambda)(3 - \lambda) - 0 = 0 [/tex]

    [tex] \lambda = 1, 3 [/tex]

    So you are right.

    To find the eigenvectors, we go back and solve this equation:

    [tex] (A - \lambda I) \vec{x} = 0 [/tex]

    for each [itex] \lambda [/itex] in turn.
     
  4. Aug 4, 2010 #3
    Yes, that's correct. The eigenvalues of a matrix A are those that satisfy the "characteristic equation"

    [tex] |\lambda \textbf{I} - \textbf{A}| = 0. [/tex]

    So for your A, we have

    [tex] (\lambda - 1)(\lambda - 3) - (0)(-2) = (\lambda - 1)(\lambda - 3) = 0. [/tex]

    So the eigenvalues of A are [tex] \lambda_1 = 1 [/tex] and [tex] \lambda_2 = 3. [/tex]

    For part (b), the eigenvectors of A are all vectors in the nullspace of [tex]\lambda \textbf{I} - \textbf{A}, [/tex] i.e., they satisfy the equationthe equation

    [tex] (\lambda \textbf{I} - \textbf{A})\vec{x} = \vec{0}. [/tex]

    EDIT: I didn't see hgfalling's post until after I'd already posted...grrr....haha. Well here's mine for what it's worth anyways.
     
  5. Aug 4, 2010 #4
    Ok, thanks for the help, but I still dont really understand the eigenvectors part of it. It would be useful if someone could write out an example. And for the A^3 bit is it the same as parts a) and b) but for AxAxA?
     
  6. Aug 4, 2010 #5
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