TITLE: The expected distortion of a linear transformation (continued)
QUESTION [5 upvotes]: Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the "average distortion" caused by the action of $A$ on vectors. 
Consider the uniform distribution on $\mathbb{S}^{n-1}$, and the random variable $X:\mathbb{S}^{n-1} \to \mathbb{R}$ defined by $X(x)=\|A(x)\|_2$. 
Question: What is the expectation of $X$? (Is there a closed formula?)
Using SVD, the problem reduces to $A$ being a diagonal matrix with non-negative entries. So, the question amounts to calculating $$\int_{\mathbb{S}^{n-1}} \sqrt{\sum_{i=1}^n (\sigma_ix_i)^2} $$ (and dividing by the volume of $\mathbb{S}^{n-1}$).

This question is related to these two, which ask about the expected distortion of the square of the norm (which is easier, since no square roots are involved).  
For the above problems, a successful approach was to use standard normal variables, in order to generate a unit random vector (see here). However, it does not seem to help in this case.

REPLY [2 votes]: Asaf,
I believe at least part of the answer you are looking for is in the paper Surface area and other measures of ellipsoids by Igor Rivin.  Look at Equation (3) to see the relationship between the quantity you seek and the ratio of the surface area and the volume of an ellipsoid.  Look at Theorem 3 that relates this ratio to an expectation of a function of Gaussian random variables.  Look at Equation (10) for an almost closed form solution, which is actually obtained from the book by A. M. Mathai and Serge B. Provost, Quadratic Forms in Random Variables (Theory and Applications).