Evaluate $\underbrace{\idotsint}_n \exp\left(-\sum_{1\le i\le j\le n}^n x_i x_j\right) \mathrm{d}x_1\cdots\mathrm{d}x_n $ I am currently studying V.I. Arnold's course, and I am stuck on this exercise:
Evaluate 
$$
\underbrace{\idotsint}_{n} 
\exp\left(-\sum_{1\le i\le j\le n}^n x_i x_j\right)
\mathrm{d}x_1\cdots\mathrm{d}x_n 
$$
Can anyone suggest a hint? Thank you in advance!
 A: Having read a lot of things by Arnold, I think his point of view was that "every student should know" the formula
$$\int\ldots\int_{\mathbb{R}^n}\exp\left\{-\frac12 x^TAx\right\}dx_1\ldots dx_n=\frac{(2\pi)^{n/2}}{\sqrt{\mathrm{det}\,A}},\tag{1}$$
where $x^T=(x_1,\ldots,x_n)$ and $A$ is a real symmetric positive definite $n\times n$ matrix. Actually, every physics student should know it indeed.
What remains is a simple linear algebra exercise.

Added: The gaussian integration formula (1) is obtained by noticing that the matrix $A$ can be brought to diagonal form by orthogonal transformation, characterized by unit jacobian. Then one is left with a product of gaussian integrals $\prod_{k=1}^n\int_{-\infty}^{\infty}\exp\{-\lambda_k x^2/2\}dx$, where $\lambda_{1\ldots n}$ denote the eigenvalues of $A$.
A: Hint: Diagonalize the symmetric matrix
$$
A=-\frac12\left(
\begin{array}{ccccc}
2&1&1&\cdots&1\\
1&2&1&\cdots&1\\
1&1&2&\cdots&1\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
1&1&1&\cdots&2
\end{array}\right)?
$$
Diagonalization of a quadratic form in the exponential = separation of variables.
