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Thread: Gaussian Integration.

  1. #1 Gaussian Integration. 
    sox
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    I'm having a bit of trouble with some basic gaussian integrals. I've not looked at this sort of stuff in ages.

    The first example on the sheet I've been given (note, this sheet isn't homework or being marked/assessed, so I can ask for help!) is the following:



    Where x is an n component vector and A is a real symmetric matrix.

    My first question is: if I take the vectors to have just one component, is the solution:



    ?

    Also, how would you go about solving the original general case?



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    Think about what you know about real symmetric matrices. I'll give you a hint: it rhymes with "Mectral Theorem"


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  4. #3  
    sox
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    So, is a diagonal matrix and...

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  5. #4  
    sox
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    Do I take the exponential and put it into form and then integrate over 1 + the sum of the diagonal elements?

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    After an orthogonal change of coordinates, you can assume that A is a diagonal matrix. At this point, the integral is simply a product of Gaussians. So you get

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    sox
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    Thanks, I think I see what you're doing there. Only bit I dont understand is why its 1 over the determinant on the bottom?

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    Think about the case when A is 1x1. For that case, you yourself supplied the answer.
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    Quote Originally Posted by sox
    Thanks, I think I see what you're doing there. Only bit I dont understand is why its 1 over the determinant on the bottom?
    Write out the problem explicitly for the case in which A is diagonal. Remember that in that case the determinant is just the product of the diagonal entries.
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  10. #9  
    sox
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    Right I get it now I think. I'll go over it later just to amke sure though. Thanks for all the help.

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  11. #10  
    sox
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    Can anyone reccomend a book that mgiht help em solve these gaussian integrals? I might see if I can get them out the library.

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  12. #11  
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    Quote Originally Posted by sox
    Can anyone reccomend a book that mgiht help em solve these gaussian integrals? I might see if I can get them out the library.
    Quantum Field Theory in a Nutshell by A. Zee talks about them quitw a bit as would most books on QFT.

    They are either pretty straightforward or purely formal. You will need to be alert to the difference since physics books tend to gloss over some significant issues (like whether or not what they are writing down is actually defined).
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