Determinantal variety and normal embedding

TL;DR

This paper proves that the intrinsic and extrinsic metrics on the space of matrices with positive determinant are bilipschitz equivalent, using the stratification by rank to analyze their geometric relationship.

Contribution

It establishes bilipschitz equivalence between intrinsic and extrinsic metrics on GL^+_n, leveraging the conical stratification structure.

Findings

Intrinsic and extrinsic metrics are bilipschitz equivalent on GL^+_n.

The conical stratification of matrix space by rank is key to the proof.

The result provides a geometric understanding of the space of matrices with positive determinant.

Abstract

The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We prove bilipschitz equivalence for intrinsic and extrinsic metrics on GL^+_n, exploiting the conical structure of the stratification of the space of n by n matrices by rank.