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Problem with integral

  1. Aug 22, 2009 #1
    What is idea for showing this?

    [tex]\int\int\ldots\int dx_1 dx_2\ldots dx_n e^{-a x^\top A x}=\sqrt{\frac{\pi^n}{\det A}}[/tex]

    [tex]A[/tex] is symmetric matrix, and [tex]a[/tex] is constant!
     
  2. jcsd
  3. Aug 22, 2009 #2
    (1) when [itex]A[/itex] is replaced by [itex]U^{-1}AU[/itex], where [tex]U[/itex] is an orthogonal matrix, the integral is unchanged.

    (2) do the case where [itex]A[/itex] is diagonal

    (3) any connection between (1) and (2) ?
     
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