Let $u_1, \cdots, u_n \in \mathbb{R}^m$ with $\|u_i\|_2 \le 1$. Let $a_1, \cdots, a_n \in \mathbb{R}_{\ge 0}$. I'm considering a sum of the form \begin{align*} S = \sum_{i=1}^{n} a_i u_i u_i^\intercal \end{align*} Note that we don't assume $(u_i)$ have any relations among themselves such as orthogonality or anything. Is there any relation we can derive between $S$ and \begin{align*} T = \sum_{i=1}^{n} \overline{a} u_i u_i^\intercal \end{align*} where $\overline{a} = n^{-1}\sum_{i=1}^{n} a_i$. As in, $S \preceq T$ or $S \succeq T$, or is neither necessarily true?
This problem is currently appearing up in a research problem where the $a_i$'s may possible be allowed to be permuted around to $a_{\sigma(i)}$, and therefore coming up with a uniform bound that is invariant under permutations, such as $\overline{a}$ would be greatly helpful.