TITLE: Expected rank of linear combination of matrices
QUESTION [8 upvotes]: Let $A_1,\dots, A_s \in M_n (\mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $\mathbb{R}$. This means that
$$ m = \min_{(c_1, ..., c_s) \in \mathbb{R}^s \backslash \{0\}} \mbox{rank} \left( \sum_{i=1}^s c_i A_i \right) > 0 $$
I am wondering about what do we expect $m$ to be greater than generically, meaning if we choose the coefficients of the matrices "randomly" what can we expect as a lower bound of $m$? Does someone see a relatively simple heuristic that gives a non-trivial expected lower bound? any input would be greatly appreciated. thank you.
This question is related to my earlier question here,
rank of a linear combination of matrices, in which I asked about the maximal size of $m$.

REPLY [11 votes]: In other words, given the "Grassmannian" $$GS_{m-1,n}=\{S\in{\bf Sym}_n\,|\,{\rm rk}\,S\le m-1\},$$
you ask what is the largest dimension $s$ of a subspace $E\subset{\bf Sym}_n$ intersecting $GS_{m-1,n}$ at the origin only, transversally (in order that this subspace be somewhat generic). This is a notoriously difficult and unsolved problem.
At least the solution when $m=2$ is very simple:
$$s=\frac{n(n+1)}2-1=\frac{(n+2)(n-1)}2.$$
Just take the subspace of trace-less symmetric matrices