I am trying to get compute at least the directional component of the following expectation, where $M$ is a symmetric, invertible, PD matrix:
$$\mathbb{E}_{v \sim N(0, I)}\left[\frac{vv^T}{||Mv||_2}\right]$$

(Note that the norm is not squared). My approach so far has been to write this as 
\begin{align}
&\mathbb{E}_{v \sim N(0, M)}\left[\frac{M^{-1/2}vv^TM^{-1/2}}{||M^{1/2}v||_2^2}\|M^{1/2}v\|_2\right] \\
&= M^{-1/2}\cdot \mathbb{E}_{v \sim N(0, M)}\left[\frac{vv^T}{||v||_M^2}\|v\|_M\right]\cdot M^{-1/2} \\
&= M^{-1/2}\cdot \mathbb{E}_{v \sim N(0, M)}\left[
\left(\frac{v}{||v||_M}\right)\left(\frac{v}{||v||_M}\right)^\top \|v\|_M\right]\cdot M^{-1/2}
\end{align}

Then, it seems as though (and supported by [this answer][1]) the first two components in the expectation should be independent of the last one, and since the norm is a scalar and the expectation of the normalized outer product is $I$, the directional component of this expectation is simply M^{-1}.

Is this correct? And if so, is there a better/more elegant way to solve this problem?


  [1]: https://mathoverflow.net/a/61736/128729