I have two sets $S_1 \subseteq S_2$ and I have a (symmetric-real) positive definite matrix $A$. I would like to know when would the following hold
$$S_1 \subseteq S_2 \Rightarrow A_{S_1} \prec A_{S_2} $$
can this be explained in terms of the Loewner ordering? Do $S_1,S_2$ need to be posets? Thank you.
Just consider for a column vector $u$, the operation $(\cdot,\cdot)$ stands for a usual scalar product: $$(A_{S_1}u,u) = (Cu,u) + \sum_{c_j\in S_1}(c_j^Tc_ju,u) = (Cu,u) + \sum_{c_j\in S_1}|c_ju|^2 \le (Cu,u) + \sum_{c_j\in S_2}|c_ju|^2 = (A_{S_1}u,u).$$ The inequality holds because $S_1\subset S_2$, hence
$$A_{S_1}\le A_{S_2}.$$