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Homework Help: Finding a Fundamental Matrix for the System

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Find a fundamental matrix for the system x'(t)=Ax(t); where
    5 -3 -2
    A = 8 -5 -4
    -4 3 3

    The second part of the question is to find eAt

    2. Relevant equations

    I know that you have to find the 3 eigenvectors and then the 'general solution' without putting the constants into the fundamental matrix. I'm having a super hard time finding the last eigen vector.

    3. The attempt at a solution

    I found the eigenvalues to be (λ-1)3 and when I completed for the first λ=1 I found that v1=[1 4 0] and v2= [1 0 2]. I tried to generalize to find v3 by using (A-λI)v3=v2 but I keep getting an invalid system.

    Any help would be great.
    Last edited by a moderator: Apr 3, 2013
  2. jcsd
  3. Apr 3, 2013 #2


    Staff: Mentor

    No, the eigenvalue is λ = 1.
    You made a mistake in v1 - it isn't an eigenvector. You can check this by verifying that Av1 ≠ 1v1.

    Your other eigenvector is correct.
  4. Apr 3, 2013 #3


    User Avatar
    Science Advisor
    Homework Helper

    And your matrix doesn't have three linearly independent eigenvectors. It only has two. If it had three it would be the identity matrix since it would be diagonalizable. I'm not sure what you mean by the 'fundamental matrix'. Can you define it?
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