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Eigen values and cubic roots question

  1. Dec 2, 2009 #1
    So I found the characteristic equation of a matrix, and I know the roots of the equation are supposed to be the eigenvalues. However, my equation is:


    I have double checked different row expansions to make sure this answer is correct. So don't worry about how I came to get that equation.
    I'm just not sure how to get roots from this. Would it be:


    So that the roots are 0 and 2?

    Basically I have trouble with cubic roots, I guess this is less of a question about eigenvalues than it is about cubic roots.

  2. jcsd
  3. Dec 2, 2009 #2


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    You are right about your example (note 0 is a double root).
    In general cubic equations are no fun. Your options are:
    1. Factor, like you did above
    2. Guess an answer [itex]\lambda_0[/itex], then divide out [itex](\lambda-\lambda_0)[/itex] and solve the resulting quadratic equation. This works well in constructed problems where you can easily see that a value like 0, 1, -1, 2 or -2 satisfies the equation.
    3. Use the analog of the quadratic formula ([itex](-b\pm\sqrt{b^2-4ac})/2a[/itex]) for cubic equations. However it is messy, and unlike the quadratic formula hardy anyone knows it by heart.
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