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Exp of a matrix

  1. Feb 12, 2014 #1
    Is correct my step by step below?

    [tex]
    \begin{aligned}
    \exp \left (\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix} \right ) &= \exp \left ( a_{11} \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} + a_{22} \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix} + a_{12} \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} + a_{21} \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \right ) \\
    & = \exp\left ( \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \right )^{a_{11}} \exp\left ( \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix} \right )^{a_{22}} \exp\left ( \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} \right )^{a_{12}} \exp\left ( \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \right )^{a_{21}} \\
    &= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{a_{11}} \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}^{a_{22}} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}^{a_{12}} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}^{a_{21}}
    \end{aligned}[/tex]
     
    Last edited by a moderator: Feb 12, 2014
  2. jcsd
  3. Feb 12, 2014 #2

    tiny-tim

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    Hi Jhenrique! :smile:

    Hint:

    i] what is exp##\begin{bmatrix} a & 0 \\ 0 & a \\ \end{bmatrix}## ? what is exp##\begin{bmatrix} 0 & a \\ a & 0 \\ \end{bmatrix}## ?

    ii] do your final matrices commute? :wink:
     
  4. Feb 12, 2014 #3

    D H

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    Jhenrique, I fixed your LaTeX so it doesn't spill across the screen.

    The answer to your question is no. You made multiple errors. Your second line erroneously assumes ##\exp(A+B) = \exp(A)\exp(B)## and also erroneously assumes ##\exp(sA)=\exp(A)^s##.
     
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