Let $n$ be a large positive integer. Let $A$ be a positive-definite matrix such with eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$ such that $\lambda_n = o(1) \to 0$ and $\lambda_i=\Theta(1)$ for all $i < n$. Let $r=(r_1,\ldots,r_n)$ be a random vector with iid components uniformly distributed on $\{\pm 1\}$ and let $g=(g_1,\dots,g_n)$ be a random vector vector with iid components distributed according to $\mathcal N(0,1)$.
Question 2. What is a good lower-bound for $\mathbb E\|Ar\|$ ?
Observation
Using Lemma 4 of this paper, it can be shown that there exist absolute constants $c,C>0$ such that
$$ c \cdot \mathbb E\|Ar\| \le \mathbb E\|Ag\| \le C\log n\cdot\mathbb E\|Ar\|. $$
On the other hand, using rotational invariance of the distribution of $g$, we have $$ \mathbb E\|Ag\| = \mathbb E[\|(\lambda_1 g_1,\ldots,\lambda_n g_n)\| \ge \mathbb E\sqrt{\sum_{i=1}^{n-1} \lambda_i^2 g_i^2} \ge \lambda_{n-1} \mathbb E\sqrt{\sum_{i=1}^{n-1} g_i^2} = \Omega(\sqrt{n}). $$ Putting things together then gives
$$ \mathbb E\|Ar\| = \Omega(\frac{\sqrt{n}}{\log n}). $$
Question 2. Can the $\log n$ factor be removed ?
Update
Note that using Jensen's inequality (and thanks to a comment by user J.), we also have the upper-bound $\mathbb E \|Ar\| \le (\mathbb E\|Ar\|^2)^{1/2} = (\mbox{trace}(A^\top A))^{1/2} = \mathcal O(\sqrt{n})$.
