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!

Simplifying a solution that has complex eigenvalues

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    I'll give an example.

    Ex: x'=[-1/2 1; -1 -1/2]x.

    2. Relevant equations

    Assume a solution of the form x=$ert for these type of problems.

    Euler's formula: ebi = cosb + isinb

    3. The attempt at a solution


    ---> r= -1/2 +/- i

    ---> x= e-t/2 ( C1(cost + isint)(1 i)T + C2(cos(-t) +isin(-t))(1 -i)T )

    I understand that I can simplify a little with the fact that sin(-t)=sin(t) and cos(-t)=-cos(t), but I don't understand how to simplify it all the way to

    C1e-t/2 (cost -sint)T + C2e-t/2(sint cost)T,

    which is the answer in the book.

    So, explain.
  2. jcsd
  3. Apr 27, 2010 #2


    User Avatar
    Homework Helper

    so basically your 2 linearly independent solutions are

    [tex] \textbf{x}_1 = e^{-t/2}(cos(t) + i.sin(t))(\begin{matrix} 1 \\ i \end{matrix}) [/tex]
    [tex] \textbf{x}_2 = e^{-t/2} (cos(-t) + i.sin(-t))(\begin{matrix} 1 \\ -i \end{matrix}) [/tex]

    note that any linear combination of these will also be a solution, so perhaps you could try taking 2 linear combinations that simplify things... making sure they are still linearly independent
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Simplifying a solution that has complex eigenvalues
  1. Complex Eigenvalues (Replies: 4)

  2. Complex Eigenvalues (Replies: 2)

  3. Complex Eigenvalues (Replies: 2)