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The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending $X_n$ locally (near $A$) to the Cartesian product of the stratum $X_{n,k}$ by the determinantal variety of complementary rank $X_{n-k}$?

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending $X_n$ locally (near $A$) to the Cartesian product of the stratum $X_{n,k}$ by the determinantal variety of complementary rank $X_{n-k}$, with reference?