Lipschitz Normal Embeddings in the Space of Matrices

TL;DR

This paper proves Lipschitz normal embeddedness for various algebraic subsets of matrix spaces, establishing bilipschitz equivalence between inner and outer metrics for these varieties.

Contribution

It demonstrates Lipschitz normal embeddedness for specific matrix varieties, including rank-constrained, symmetric, skew-symmetric, and triangular matrices, and discusses potential generalizations.

Findings

Lipschitz normal embeddedness of matrix spaces and their closures.

Bilipschitz equivalence between inner and outer metrics.

Discussion on extending results to determinantal varieties.

Abstract

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety 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 $m \times n$ matrices, symmetric matrices and skew-symmetric matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant $0$. We also make a short discussion about generalizing these results to determinantal varieties in real and complex spaces.