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Homework Help: Matrix Exponential

  1. Apr 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Given x' = Ax where A =

    ( 0 1 )
    ( -1 0 )

    Compute the matrix exponential and then find the solution such that x(0) =

    ( 1 )
    ( 2 )

    2. Relevant equations

    3. The attempt at a solution

    I computed the matrix exponential and obtained the matrix,

    e^(A) =

    ( cos(t) sin(t) )
    ( -sin(t) cos(t) )

    But I don't understand how to compute the initial condition. Am I supposed to compute the initial by multiplying the original A by x(0) and then compute the matrix exponential for the new A? Or multiple e^(A) by x(0)? My notes aren't very clear. But those are my only guesses..

    Thanks for any help.
    Last edited: Apr 26, 2009
  2. jcsd
  3. Apr 26, 2009 #2


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    Science Advisor
    Homework Helper

    What you computed was e^(At). And sure, your solution is then x(t)=e^(At)x(0). If you take d/dt of that then you get x'(t)=Ax(t), right?
  4. Apr 26, 2009 #3
    Alright, I think I've got it. Compute x(t) then differentiate it?
  5. Apr 26, 2009 #4


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    You don't have to differentiate it, I was just pointing out why x(t)=e^(At)x(0) works as a solution. x'(t)=Ax(t) and x(0)=e^(0)x(0). It satisfies the ode and has the right initial condition.
  6. Apr 26, 2009 #5
    Alright, thank you.
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