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Homework Help: Please help with this matrix question! Solve for Eigenvales, and Eigenvectors

  1. Feb 12, 2012 #1
    M = (a c)
    (c b)

    Sorry for the double sets of brackets, its all in one. I'll also show as far as i got below:

    [a-λ c] => (a-λ)(b-λ) - c^2 = λ^2 + (-a-b)λ + (ab-c^2) =0
    [c b-λ] =>

    then using the quadratic formula: λ = [-(-a-b) +/- Sqrt{(-a-b)^2 - 4(1)(ab-c^2)}]/ 2

    then after some algebra I got stuck: λ = [(a+b) +/- Sqrt{a^2 + 2ab + b^2 -4ac^2}]/2

    λ = [(a+b) +/- Sqrt{a^2 - 2ab + b^2 + c^2}]/2 ==>> THIS IS WHERE I GOT STUCK. PLEASE HELP
     
  2. jcsd
  3. Feb 12, 2012 #2

    tiny-tim

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    welcome to pf!

    hi sidinsky! welcome to pf! :wink:

    (have a square-root: √ and a ± and try using the X2 button just above the Reply box :wink:)
    looks ok :smile:

    that gives you the two eigenvalues,

    so now use the standard techniques to find an eigenvector for each :wink:
     
  4. Feb 13, 2012 #3
    is there any way this expression can be simplified further? I am sorta having trouble with the algebra :S
     
  5. Feb 14, 2012 #4

    tiny-tim

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    i don't think so

    how far have you got?​
     
  6. Feb 14, 2012 #5
    well I managed to clean up the expression inside the sqrt a bit to: [(a-b)2 + c 2 ]/2

    a2 -2ab + b2 + c2 = (a-b)2 + c2

    λ1 = (a+b) + sqrt{(a-b)2 +c 2}[itex]/2[/itex]

    and λ2 = (a+b) - sqrt{(a-b)2 + c2}[itex]/2[/itex]

    now these expressions for Lambda are too difficult for me to use to solve for eigenvectors
     
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