What is the expected value of the norm of a random vector with Gaussian distribution?

Question

Let's assume I have a zero-mean Gaussian random vector $x$ with co-variance matrix $\Sigma$. Is there a closed form equation for $\mathbb E[||x||]= \mathbb E[\sqrt{x^T x}]$?

Answer 1

The usual expression is a finite number of terms involving the variances $(\sigma_i)$ and the co-variances $(c_{ij})$. $E||x||^2=\sum_i \sigma_i^2+\sum_i\sum_{j\ne i}c_{ij}$.