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Complex Eigenvalues

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Apply the eigenvalue method to find a general solution of the given system.
    [tex]x_1' = 5x_1 - 9x_2[/tex]
    [tex]x_2' = 2x_1 - x_2[/tex]


    2. Relevant equations
    (A-λI)v=0


    3. The attempt at a solution
    [tex]x_1' = 5x_1 - 9x_2[/tex]
    [tex]x_2' = 2x_1 - x_2[/tex]

    [tex]\left[
    \begin{array}{cc}
    5-λ & -9\\
    2 & -1-λ
    \end{array}
    \right]=(5-λ)(-1-λ)+18=0[/tex]
    [tex]λ^2-4λ-13=0[/tex]
    [tex](λ-2)^2 -9=0[/tex]
    [tex]λ=2+3i,\overline{λ}=2-3i[/tex]
    So I plugged λ into the matrix;
    [tex]\left[
    \begin{array}{cc}
    3-3i & -9\\
    2 & -3-3i
    \end{array}
    \right]
    \left[
    \begin{array}{cc}
    a\\
    b
    \end{array}
    \right]=(3-3i)a-9b=0
    2a-(3+3i)b=0[/tex]
    This is where I'm stuck...the answer is;
    [tex]x_1(t)=3e^{2t}(c_1cos2t - 5c_2sin2t)[/tex]
    [tex]x_2(t)=e^{2t}[(c_1+c_2)cos3t + (c_1-c_2)sin3t)][/tex]
     
  2. jcsd
  3. Mar 21, 2012 #2
    Any one?
     
  4. Mar 21, 2012 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    This is your error. [itex](\lambda- 2)^2- 9= 0[/itex] is equivalent to [itex](\lambda- 2)^2= 9[/itex] so [itex]\lambda- 2= \pm 3[/itex]. There is no "i".

    There are no eigenvectors because 2+ 3i and 2- 3i are not eigenvalues.
     
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