Let $X= \{A_i \in \mathbb{C}^{n \times n} : i \in \mathbb{N}\}$ be a set of rank 1 matrices with the property that for any $S \subset \mathbb{N}$ of size $n$, $\sum_{i \in S} A_i$ is invertible. What is the minimum dimension of $\text{span}(X)$?
Some basic observations: it's at least $n$, since otherwise you could write the sum of any $n$ of the $A_i$'s as a linear combination of some $n-1$ of the $A_i$'s. It's at most $n^2$, since any random choice of $A_i$'s has this property and spans a space of dimension at most $n^2$. You can improve this upper bound to $2n-1$ by taking $A_i = (x_i, x_i^2, \ldots, x_i^n)^T(x_i, x_i^2, \ldots, x_i^n)$ for some distinct $x_i \in \mathbb{C}$. The sum of any $n$ matrices of this form can be written as a product $VV^T$ for some invertible Vandermonde matrix $V$. The $A_i$'s span a subspace of dimension $2n-1$ in this case (the subspace of Hankel matrices).
Is $2n-1$ the smallest you can achieve?
(IIRC there is a simple proof that this is true when you restrict the $A_i$'s to be symmetric, but I forget it...)
