1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook