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 [tex]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) ?[/tex]