Lipschitz normal embeddings in the space of matrices

A semi-algebraic subset in Rⁿ or Cⁿ is naturally equipped with two different metrics, the inner metric and the outer metric. Such a set (or its germ) is called Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space of rectangular/(skew-)symmetric/hermitian matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant 0. (In these cases we establish explicit bilipschitz constants.) We also make a short discussion about generalizing these results to determinantal varieties in real and complex spaces.