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Homework Help: Exponential matrix problem and Putzer's formula

  1. Jun 18, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]e^{ \begin{bmatrix} \lambda_{1} & 0 \\ 1& \lambda_{2} \end{bmatrix}t } [/itex]=\begin{bmatrix} e^{\lambda_{1}t} & 0 \\
    \frac{e^{\lambda_{2}t} - e^{\lambda_{1}t}}{ \lambda_{2} - \lambda_{1}} & e^{\lambda_{2}t}
    \end{bmatrix}

    with [itex]\lambda_{1}\neq \lambda_{2}[/itex]
    2. Relevant equations

    Could someone solve this for me?

    3. The attempt at a solution

    I am no good at maths... with basic knowledge of linear algebra.
    It looks like [itex]\lambda_{1} , \lambda_{2}[/itex] are the eigenvalues of a matrix A that solves a differential system. All indications are pointing to Putzer's formula, but everything i have tried failed. Probably i am missing something....

    Thanks in advance...
    p.s.Sorry for my terrible English
     
  2. jcsd
  3. Jun 18, 2011 #2

    hunt_mat

    User Avatar
    Homework Helper

    Basically you write out the definition of the exponential:
    [tex]
    e^{x}=1+x+\frac{x^{2}}{2!}+\cdots +\frac{x^{n}}{n!}+\cdots
    [/tex]
    To compute the LHS of your equation, you will have you figure out an equation for:
    [tex]
    \left(
    \begin{array}{cc}
    \lambda_{1} & 0 \\
    1 & \lambda_{2}
    \end{array}\right)^{n}
    [/tex]
    and this will give you what you require.
     
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