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Simple matrix models for the flag, Grassmann, and Stiefel manifolds

Lek-Heng Lim, Ke Ye

TL;DR

This work constructs and classifies orthogonally-equivariant, low-dimensional matrix representations of the Grassmannian, flag, and Stiefel manifolds. It introduces two complementary families: isospectral (and traceless) models for Grassmannians/flags and quadratic variants, and Cholesky models for Stiefel manifolds, proving their exhaustiveness under the stated equivariance and minimal-dimension criteria. The authors establish explicit diffeomorphisms between parameterizations, derive invariant Riemannian metrics aligned with the bi-invariant metric on SO($n$), and demonstrate that these models yield closed-form matrix expressions for geometric objects, enabling stable, efficient numerical computations. The results offer flexible, condition-number-aware options for practical computations on these manifolds and reveal deep connections to Cartan geometry via the positive definite cone for the Stiefel case.

Abstract

We derive three families of orthogonally-equivariant matrix submanifold models for the Grassmann, flag, and Stiefel manifolds respectively. These families are exhaustive -- every orthogonally-equivariant submanifold model of the lowest dimension for any of these manifolds is necessarily a member of the respective family, with a small number of exceptions. They have several computationally desirable features. The orthogonal equivariance allows one to obtain, for various differential geometric objects and operations, closed-form analytic expressions that are readily computable with standard numerical linear algebra. The minimal dimension aspect translates directly to a speed advantage in computations. And having an exhaustive list of all possible matrix models permits one to identify the model with the lowest matrix condition number, which translates to an accuracy advantage in computations. As an interesting aside, we will see that the family of models for the Stiefel manifold is naturally parameterized by the Cartan manifold, i.e., the positive definite cone equipped with its natural Riemannian metric.

Simple matrix models for the flag, Grassmann, and Stiefel manifolds

TL;DR

This work constructs and classifies orthogonally-equivariant, low-dimensional matrix representations of the Grassmannian, flag, and Stiefel manifolds. It introduces two complementary families: isospectral (and traceless) models for Grassmannians/flags and quadratic variants, and Cholesky models for Stiefel manifolds, proving their exhaustiveness under the stated equivariance and minimal-dimension criteria. The authors establish explicit diffeomorphisms between parameterizations, derive invariant Riemannian metrics aligned with the bi-invariant metric on SO(), and demonstrate that these models yield closed-form matrix expressions for geometric objects, enabling stable, efficient numerical computations. The results offer flexible, condition-number-aware options for practical computations on these manifolds and reveal deep connections to Cartan geometry via the positive definite cone for the Stiefel case.

Abstract

We derive three families of orthogonally-equivariant matrix submanifold models for the Grassmann, flag, and Stiefel manifolds respectively. These families are exhaustive -- every orthogonally-equivariant submanifold model of the lowest dimension for any of these manifolds is necessarily a member of the respective family, with a small number of exceptions. They have several computationally desirable features. The orthogonal equivariance allows one to obtain, for various differential geometric objects and operations, closed-form analytic expressions that are readily computable with standard numerical linear algebra. The minimal dimension aspect translates directly to a speed advantage in computations. And having an exhaustive list of all possible matrix models permits one to identify the model with the lowest matrix condition number, which translates to an accuracy advantage in computations. As an interesting aside, we will see that the family of models for the Stiefel manifold is naturally parameterized by the Cartan manifold, i.e., the positive definite cone equipped with its natural Riemannian metric.
Paper Structure (15 sections, 17 theorems, 87 equations)

This paper contains 15 sections, 17 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Computational significance
  3. Contributions
  4. Notations and some background
  5. Linear algebra
  6. Differential geometry
  7. Existing models of the flag manifold
  8. Equivariant matrix models of the Grassmannian and flag manifold
  9. Change-of-coordinates for isospectral models
  10. Equivariant matrix models for Stiefel manifolds
  11. Equivariant Riemannian metrics
  12. Quadratic model of the Grassmannian
  13. Cholesky model of the Stiefel manifold
  14. Isospectral model of the flag manifold
  15. Conclusion

Key Result

Lemma 3.1

Let $0 \eqqcolon k_0 < k_1 < \cdots < k_p < k_{p+1} \coloneqq n$ be integers and Then the maps below are all diffeomorphisms: \begin{tikzcd} {\Flag(k_1,\dots, k_p,\mathbb{R}^n)} && {\Flag_{\Gr}(n_1,\dots,n_{p+1})} \\ {\GL_{n}(\mathbb{R})/\P_{k_1,\dots, k_p}(\mathbb{R})} && {\SO_{n}(\mathbb{R})/\S(\O_{n_1}(\mathbb{R}) \times \cdots \times \O_{n_{p+1}}(\mathbb{R}) )} \arrow[" where for ea

Theorems & Definitions (33)

  • Definition 2.1: Equivariant embedding and equivariant submanifold
  • Lemma 3.1: Change-of-coordinates for flag manifolds I
  • proof
  • Theorem 4.1: Isospectral model I
  • proof
  • Corollary 4.2: Isospectral model II
  • Corollary 4.3: Traceless isospectral model
  • proof
  • Corollary 4.4: Quadratic model
  • Proposition 4.5: Simpler isospectral model
  • ...and 23 more