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Consider the collection of $n$ by $n$ matrices $$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$ where $c_i\in \{0,1\}$, $P_i$ and $Q_i$ are disjoint index sets, and $A(P_i;Q_i)$ is the submatrix formed by taking just the rows $P_i$ and columns $Q_i$ of $A$.

What kind of conditions on $S$ would be natural to add to guarantee that $S$ is path connected? Also what areas of mathematics are useful in trying to answer these types of path connected questions?

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Set of non-positive matrices in general position. I would look at realisable oriented matroids. The chapter in… addresses these things – Rabee Tourky Jun 21 at 4:54

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