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Math Help - Computing e^A

  1. #1
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    Computing e^A

    I need to compute e^A for the matrix
    A= 0 ∏
    ....-∏ 0

    where the diagonal are zeros and the other diagonal has pi on top and negative pi on the bottom.
    I'm not quite sure where to start.

    Thanks
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  2. #2
    Senior Member vincisonfire's Avatar
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     e^A is defined as
    *\sum_{i=0}^{\infty} \frac{A^n}{n!}
    You can find that
    if  B=e^A
    then  B(1,1) = B(2,2) =1 -\frac{\pi^2}{2!}+ \frac{\pi^4}{4!} - \frac{\pi^6}{6!} + \frac{\pi^8}{8!} - ... =-1
     B(1,2) = \frac{\pi}{1!}- \frac{\pi^3}{3!} + \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + ... = 0
     B(2,1) = -\frac{\pi}{1!}+ \frac{\pi^3}{3!} - \frac{\pi^5}{5!} + \frac{\pi^7}{7!} - ... = 0
    If your prof wants you to evaluate those series then I don't know (I used Maple). Well I don't have the time... or both.
    Last edited by vincisonfire; December 1st 2008 at 04:21 AM.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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     B(1,1) = B(2,2) should be  -1
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  4. #4
    Senior Member vincisonfire's Avatar
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    I don't think so because it is
     \sum_{i=1}^{\infty} (-1)^{n}\cdot\frac{\pi^{2n}}{(2n)!} because the first term is 0 not 1 we must integrate from 1 to infinity not 0 to infinity. But I may be mistaking.
    Here are the 10 first terms
    -4.934802202
    -0.876090073
    -2.211352843
    -1.976022212
    -2.001829103
    -1.999899529
    -2.000004167
    -1.999999864
    -2.000000003
    -1.999999999
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    well I typed e^A in my ti-89 and it gave me back  -I where I is the identity matrix
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  6. #6
    Senior Member vincisonfire's Avatar
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    Yes you're right because we have to add the identity matrix at the beginning. SOrry.
     B=e^A
    then  B(1,1) = B(2,2) =1 -\frac{\pi^2}{2!}+ \frac{\pi^4}{4!} - \frac{\pi^6}{6!} + \frac{\pi^8}{8!} - ... =-1
     B(1,2) = \frac{\pi}{1!}- \frac{\pi^3}{3!} + \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + ... = 0
     B(2,1) = -\frac{\pi}{1!}+ \frac{\pi^3}{3!} - \frac{\pi^5}{5!} + \frac{\pi^7}{7!} - ... = 0
    Somebody knows how to calculate these sum by hand?
    Last edited by vincisonfire; November 27th 2008 at 04:05 AM.
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  7. #7
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    Quote Originally Posted by victor1487 View Post
    I need to compute e^A for the matrix
    A= 0 ∏
    ....-∏ 0

    where the diagonal are zeros and the other diagonal has pi on top and negative pi on the bottom.
    I'm not quite sure where to start.

    Thanks
    You might consider starting by diagonalizing A--

    Notice that A = P D P^{-1}

    where

    P = \begin{pmatrix}1 &1 \\<br />
i &-i \end{pmatrix}

    D =\begin{pmatrix}\pi i &0 \\<br />
0 &-\pi i \end{pmatrix}

    P^{-1} = \frac{1}{2} \begin{pmatrix}1 &-i \\<br />
1 &i \end{pmatrix}

    (I'm assuming you know how to go about diagonalizing a matrix; if not, there is an article on Wikipedia:

    Diagonalizable matrix - Wikipedia, the free encyclopedia.)
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