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Homework Help: Trying to follow my textbook's explanation

  1. Apr 21, 2010 #1
    I don't need the template because this isn't a homework problem, per se; it's just information about how to get started on my homework.

    So I have an equation x'=Ax, x(0)=x0, where A is a constant matrix.

    I can write it as x=ø(t)x0, where ø(t) is a fundamental matrix such that ø(0)=I

    We know that Taylor expanding eat gives us ∑ antn/n!

    (n starts from 1 and goes to infinity)

    I + ∑Antn / n! = I + At + A2t2/2! + ..... + Antn/n! + ........ = e(At)

    So, I don't even understand how you can raise a scalar to a power of a matrix. This is a messed-up world we live in.

    d/dt e(At) = ∑Antn-1/(n-1)! = A[ I + ∑Antn/n!].

    I don't understand that last step. Must be some property of summations?
     
  2. jcsd
  3. Apr 21, 2010 #2
    write out some terms (say the first 3) instead of just ∑...

    it should become clear.
     
  4. Apr 21, 2010 #3

    HallsofIvy

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    Science Advisor

    There is no "scalar to a power of a matrix" except on the far right. And that equation defines what is meant by [itex]e^A[/itex]

    [itex]\sum_{n=0}^\infty A^n t^{n-1}=[/itex][itex] A+ A^2t+ A^3t^2+ \cdot\cdot\cdot=[/itex][itex] A(I+ At+ A^2 t^2+ \cdot\cdot\cdot)= A(I+ \sum_{n=1}^\infty A^nt^n)[/itex].
     
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