Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^*$ which is obtained from the first one plus some injected noise $\eta$, where every entry $\eta_{i,j}$ is i.i.d. and comes from a normal distribution with mean $0$ and variance such that $||\eta||_F = 1/10$. In other words, as Dirk pointed out in the answer,the entries of η (viewed vectorized) come from the uniform distribution on the scaled sphere in $n^2$ dimensions.

What can we say about the following expected value $$\mathbb{E}[| Mx - M^*x|]?$$

I know that $\mathbb{E}|\eta(x)| <1/10$ since, because of the frobenius norm of the noise $|\eta(x)| < 1/10$, so the equality only holds when vector $x$ is aligned with the highest eigenvector of the noise matrix $\eta$, but this happens with low probability, and this probability should (intuitively) change with respect to $m$: in a higher dimensional space the probability of two random vectors having a similar direction is lower than in a 2-dimensional one.

Is there a way to improve the bound on this expected value?

notiid andnotnormal. $\endgroup$ – Robert Israel Jan 14 at 21:12