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Description
Let $A$ be an $m \times n$ real matrix. If the manifolds ${\widetilde{\cal M}_A}= \{ H^{-1} A G : G, H \text{ are nonsingular} \}$ and $Q(\text{sgn}(A))$ intersect transversally at $A,$ that is, the tangent spaces of ${\widetilde{\cal M}_A}$ and $Q(\text{sgn}(A))$ at $A$ sum to $ \mathbb R ^{m\times n},$ we say that $A$ has the rank-preserving transversality property (RPTP) and that $A$ is an RPTP matrix. We establish many important properties of RPTP matrices as well as several sign pattern classes that require the RPTP. For example, RPTP matrices are closed under permutation equivalence, diagonal equivalence, and transpose. Further, a block upper triangular matrix with all diagonal blocks square has the RPTP if and only if each diagonal block has the RPTP and at most one diagonal block is singular. Just as the Strong Spectral Property is useful in studying the spectra of symmetric matrices associated with a graph, the notion of RPTP is a useful tool for studying the minimum ranks of sign patterns and zero-nonzero patterns.