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Let $T\colon\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^{n\times n}$ and suppose that $\text{rank} \ T(I_n) = n$. Under what conditions is it true that there exists a neighborhood $N_I$ of the identity such that, for all $A\in N_I$, $\text{rank} \ T(A) = n$?

Can stronger conditions be provided in the case where the range of the function is a subset of the positive semidefinite matrices of dimension $n$?

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$T$ being continuous at $I_n$ is sufficient.

All $(n-1)\times(n-1)$ minors of $T(I_n)$ are non-zero, minors are continuous functions, and there are only finitely many of them, meaning there is some $\varepsilon>0$ such that we have an open ball of radius $\varepsilon$ around $T(I_n)$ with only invertible matrices. By continuity of $T$, there is a $\delta>0$ such that setting $N_I$ to be the open $\delta$-ball around $I$ works.

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