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Systems of ODE's - Complex Eigenvalues

  1. Apr 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the general solution of the given system.

    The given matrix is X' = (1st row (1,-1,2) 2nd row (-1,1,0) 3rd row (-1,0,1))X

    2. The attempt at a solution

    The eigenvalue determinant = (1st row (1-λ,-1,2) second row (-1,1-λ,0) 3rd row (-1,0,1-λ)

    Solving the determinant gives -(λ-1)(λ^2-2λ+2)

    So λ1,2 = 1+i and 1-i and λ3 = 1

    Now my question is how do I solve the 3x3 matrix with copmlex numbers in it? I used gaussian elimination to solve for the real value λ3.

    (1st row (-i,-1,2) 2nd row (-1,-i,0) 3rd row (-1,0,-i)) multiplied by the column (k1,k2,k3) = 0
     
    Last edited: Apr 28, 2010
  2. jcsd
  3. Apr 28, 2010 #2

    gabbagabbahey

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    Your two complex eigenvalues have a sign error. The roots of [itex]\lambda^2-2\lambda+2=0[/itex] are [itex]\lambda=1\pm i[/itex].

    You can still use Gaussian elimination to solve for the eigenvectors.
     
  4. Apr 28, 2010 #3
    Can someone show me how?
     
  5. Apr 28, 2010 #4

    gabbagabbahey

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    You do it exactly the same way you did it for your real eigenvalue....try it out and post your work if you get stuck.
     
  6. Apr 28, 2010 #5
    Is multiplying a row by a complex number a valid operation?
     
  7. Apr 28, 2010 #6

    gabbagabbahey

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    Yes. It's exactly the same as multiplying both sides of an equation by a complex number.
     
  8. Apr 28, 2010 #7
    The corrected eigenvalue matrix is
    [-i,-1,2]
    [-1,-i,0]
    [-1,0,-i]

    using the operations 1) i*R1, 2) R1+R2, 3) R1+R3, 4) (-1/2)R2+R3, 5) -2*R2+R1 gives

    [1,-i,0]
    [0,-i, i]
    [0,0,0]

    which says k2=k3; picking k1=1 and k2=k3= i gives one solution,k=
    [1]

    which is split into the real and imaginary parts B1 and B2, respectively

    [1]
    [0]
    [0]

    [0]
    [1]
    [1]

    The general solution is c1[B1*cos(t)-B2sin(t)]e^t + c2[B2*cos(t)-B1sin(t)]e^t +c3
    ([0])e^t
    ([1])
    ([2])
     
    Last edited: Apr 28, 2010
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