Solution for Integral Probs: Exponential Form w/ Symmetric Matrix

[Petar Mali]

What is idea for showing this?

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

A is symmetric matrix, and a is constant!

 

[g_edgar]

(1) when A is replaced by U^{-1}AU, where U[/itex] is an orthogonal matrix, the integral is unchanged.<br /> <br /> (2) do the case where A is diagonal<br /> <br /> (3) any connection between (1) and (2) ?