<p>I would like to ask the following question:</p>

<blockquote>
  <p>if <span class="math-container">$M$</span> is a <span class="math-container">$m\times n$</span> constant matrix and <span class="math-container">$\eta\sim\mathcal{N}(0,I)$</span>, then does <span class="math-container">$$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M\eta\rVert}{\lVert\eta\rVert}\right]$$</span>
  exist? Also, let <span class="math-container">$x\in \mathbb{R}^n_{\ne 0}$</span> be an arbitrary non-zero vector. Is it possible to compute the maximum (or at least to find a tight upper-bound) over all <span class="math-container">$x$</span>, of the quantity <span class="math-container">$$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M(x+\lVert x\rVert \eta)-Mx\rVert}{\lVert Mx \rVert}\right]=\lVert x\rVert\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M \eta\rVert}{\lVert Mx \rVert}\right]$$</span></p>
</blockquote>

<p>Should I ask here or on Mathematics?</p>
