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Finding eigenvalues, Shankar exercise 1.8.3

  1. Oct 25, 2010 #1
    First, I appologise if this is in the wrong place, while the book is QM, the question is pure maths. Also I'm not sure if this techically counts as homework as I am self studying. Finally, sorry for the poor formatting, I'm not that good with LaTeX

    1. The problem statement, all variables and given/known data

    Given the matrix: [tex]\Omega[/tex] =
    [tex]\left[ {\begin{array}{ccc}
    2 & 0 & 0 \\
    0 & 3 & -1 \\
    0 & -1 & 3 \\
    \end{array} } \right]
    [/tex]

    Show that [tex]\omega[/tex]1 = [tex]\omega[/tex]2 = 1; [tex]\omega[/tex]3 = 2

    2. Relevant equations



    3. The attempt at a solution

    So det([tex]\Omega[/tex] - [tex]\omega[/tex]I) = (2 - [tex]\omega[/tex])((3 - [tex]\omega[/tex])(3 - [tex]\omega[/tex]) - 1)

    Which obviously leaves [tex]\omega[/tex] = 2, but also (3 - [tex]\omega[/tex])2 = 1, the solutions to which should be [tex]\omega[/tex] = 2 and [tex]\omega[/tex] = 4.

    Where am I going wrong?

    Any help greatfully appreciated.
     
  2. jcsd
  3. Oct 25, 2010 #2

    vela

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    You're not going wrong for the given matrix. Are you sure you're doing the right problem?
     
  4. Oct 26, 2010 #3
    You're absolutely right, there was a factor of a half in the original question that I completely missed, thanks. I must have checked the original problem a dozen times before posting and didn't spot it, I hate my brain sometimes.

    Thanks again,

    Tim
     
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