Let $A$ be an $n\times n$ real matrix, if the manifolds ${\cal M}_A=\{ G^{-1}AG|G\in\text{GL}(n,\mathbb R)\}$ and $Q(\text{sgn}(A))$ (consisting of all real matrices having the same sign pattern as $A$), both considered as embedded submanifolds of $\mathbb R^{n \times n}$, intersect transversally at $A$, then every superpattern of sgn$(A)$ also allows a matrix similar to $A$. In this paper, we define this similarity-transversality property (STP) as the full row rank property of the Jacobian matrix of the entries of $AX-XA$ at the zero entry positions of $A$ with respect to the nondiagonal entries of $X$, where $X=[x_{ij}]$ is a generic matrix of order $n$ whose entries are independent variables. This approach makes it possible to take better advantage of the combinatorial structure of the matrix $A$, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. We provide some examples to illustrate possible applications.

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Last modified: Tue Apr 9 10:59:23 EDT 2024