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 $\eta_{i,j}$ comes from a Gaussian distribution with mean $0$, such that $||\eta||_F = 1/10$.
What can we say about the following expected value $$\mathbb{E}[| Mx - M^*x|]?$$
EDIT: How it has been suggested in an answer, $\mathbb{E}|\eta(x)| <1/10$ since $|\eta(x)| < 1/10$ because of the Frobenius norm of the noise, 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?