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Evaluated e^At

  1. Nov 6, 2009 #1
    I did a problem in class today that evaluated [tex]f(t)=e^{At}[/tex] for [tex]A_{2,2}=\begin{bmatrix}2&1 \\-1&4 \end{bmatrix}[/tex] to a matrix form.

    The answer I got was:

    [tex]f(t)=\begin{bmatrix}e^{3t}-te^{3t}&te^{3t} \\-te^{3t}&e^{3t}+te^{3t} \end{bmatrix}[/tex]

    Factoring we have:

    [tex]f(t)=e^{3t}\begin{bmatrix}1-t&t \\-t&1+t \end{bmatrix}[/tex]

    My question is, is there some simple general expression for simplifying [tex]e^{At}[/tex] to a matrix form? Maybe something that resembles [tex]e^{tA_{2,2}}=e^{\lambda t}\begin{bmatrix}1-t&t \\-t&1+t \end{bmatrix}[/tex]

    but for any size matrix.
     
  2. jcsd
  3. Nov 6, 2009 #2

    tiny-tim

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    Hi epkid08! :wink:

    If you can write A in the form PQP-1 where Q is diagonal …

    then ∑ An/n! = P(∑ Qn/n!)P-1 = PeQP-1, where eQ = … ? :smile:

    (oh, and your simple form with a single exponential factor on the outside only works in thsi case because there is a double eigenvalue :wink:)
     
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