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Homework Help: Discriminant of Characteristic Polynomial > 0

  1. Nov 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that the descriminant of the characteristic polynomial of K is greater than 0.

    [tex]K=\begin{pmatrix}-k_{01}-k_{21} & k_{12}\\
    k_{21} & -k_{12}

    And [itex]k_i > 0[/itex]

    2. Relevant equations


    3. The attempt at a solution

    I have tried the following:
    \begin{pmatrix}-k_{01}-k_{21}-\lambda & k_{12}\\
    k_{21} & -k_{12}-\lambda

    Bringing me to

    And then plugging it into discriminant form


    But from there I don't think that is a true statement.

    Any help would be appreciated, thanks.
  2. jcsd
  3. Nov 26, 2012 #2


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    Science Advisor
    Homework Helper

    Actually, I think it is true. But it's not obvious. Let's call k12=x, k01=y and k21=z, so you want to show (x+y+z)^2-4yz>0 if x>0, y>0 and z>0. Just so we don't have to write the subscripts. I showed it by completing as many squares as I could in that expression after expanding it. Then it's easy to see.
    Last edited: Nov 26, 2012
  4. Nov 26, 2012 #3
    D'oh I think the form I was looking for was:


    which is clearly greater than zero.

    Thanks for the insight.
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