Rank-constrained linear combination of matrices

Question

I have several real symmetric matrices, say, $A_1, A_2,\dots, A_n$. I want to know if there exists a complex linear combination of these matrices

$$M := \sum_{i=1}^n c_i A_i , \quad c_i \in \mathbb{C}$$

satisfying $\text{rank}(M)\leq2$. If there are such $c_i$'s, what are they? Is there any method of doing this?

Answer 1

You want to find $(c_1, \ldots, c_n)$ such that the $3 \times 3$ minors of $M$ are all $0$. Each of these minors is a polynomial in $c_1, \ldots, c_n$.
That may be rather a lot of minors, so I would start with maybe $n$ of them, find the solutions, and then see which make $M$ have rank $\le 2$.