# Principal bundle structure of matrix manifolds

Marie Billaud-Friess, Antonio Falco, Anthony Nouy

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Marie Billaud-Friess, Antonio Falco, Anthony Nouy. Principal bundle structure of matrix manifolds. Mathematics, 2021, 9 (14), pp.1669. 10.3390/math9141669. hal-01521748

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Principal Bundle Structure of Matrix Manifolds

Marie Billaud-Friess <fnref n="1" /> ID, Antonio Falcó <fnref n="2" />,<fnref n="3" />,* ID and Anthony Nouy <fnref n="1" /> ID

**Abstract:** In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r < k$ in $\mathbb{R}^k$, which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on $\mathbb{R}^{(k-r) \times r}$. Then, we define an atlas for the set $M_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $GL_r$, the general linear group of invertible matrices in $\mathbb{R}^{k \times k}$. Finally, we define an atlas for the set $M_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $GL_r$. The atlas of $M_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $M_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $M_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

**Keywords:** matrix manifolds; low-rank matrices; Grassmann manifold; principal bundles

check for
updates

**Citation:** Billaud-Friess, M.; Falcó, A.; Nouy, A. Principal Bundle Structure of Matrix Manifolds. Mathematics 2021, 9, 1669. https://doi.org/10.3390/math9141669

Academic Editor: Marian Ioan Munteanu

Received: 11 June 2021
Accepted: 14 July 2021
Published: 15 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

<figure><img src="image_1.png" /></figure>

**Copyright:** © 2021 by the authors. License MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

# 1. Introduction

Low-rank matrices appear in many applications involving high-dimensional data. Low-rank models are commonly used in statistics, machine learning or data analysis (see [1] for a recent survey). Furthermore, low-rank approximation of matrices is the cornerstone of many modern numerical methods for high-dimensional problems in computational science, such as model-order-reduction methods for dynamical systems or parameter-dependent or stochastic equations [2–5].

These applications yield problems of approximation or optimization in the sets of matrices with fixed rank:

$$M_r(\mathbb{R}^{n \times m}) = \{Z \in \mathbb{R}^{n \times m} : \mathrm{rank}(Z) = r\}.$$

Fixed-rank matrices appear also in the theory of characteristics of Partial Differential Equations and Monge-Ampère equations [6]. More precisely, it has been proven [6,7] that Monge-Ampère equations with $n$ independent variables and of Goursat-type are in one-to-one correspondence with the set $\{Z \in M_r(\mathbb{R}^{n \times n}) : r \le 2\}$. Thus, the parabolic or hyperbolic nature of the Monge-Ampère equation is related to the rank of such matrices.

In [8,9], the authors point out that Algebraic Geometry appears as a natural tool in study of the set $M_r(\mathbb{R}^{n \times m})$. We wish to mention the papers [10–12] that raise the natural question of how large these matrix spaces are.

<footnote n="1">¹ Department of Computer Science and Mathematics, Ecole Centrale de Nantes, 1 Rue de la Noë, BP 92101, CEDEX 3, 44321 Nantes, France; marie.billaud-friess@ec-nantes.fr (M.B.-F.); anthony.nouy@ec-nantes.fr (A.N.)</footnote>

<footnote n="2">² Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad CEU Cardenal Herrera, CEU Universities, San Bartolomé 55, 46115 Alfara del Patriarca, Spain</footnote>

<footnote n="3">³ ESI International Chair@CEU-UCH, Universidad Cardenal Herrera-CEU, CEU Universities San Bartolomé 55, 46115 Alfara del Patriarca, Spain</footnote>

<footnote>* Correspondence: afalco@uchceu.es</footnote>
Principal Bundle Structure of Matrix Manifolds

Marie Billaud-Friess <fnref n="1" /> ID, Antonio Falcó <fnref n="2" />,<fnref n="3" />,* ID and Anthony Nouy <fnref n="1" /> ID

**Abstract:** In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r < k$ in $\mathbb{R}^k$, which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on $\mathbb{R}^{(k-r) \times r}$. Then, we define an atlas for the set $M_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $GL_r$, the general linear group of invertible matrices in $\mathbb{R}^{k \times k}$. Finally, we define an atlas for the set $M_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $GL_r$. The atlas of $M_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $M_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $M_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

**Keywords:** matrix manifolds; low-rank matrices; Grassmann manifold; principal bundles

check for
updates

**Citation:** Billaud-Friess, M.; Falcó, A.; Nouy, A. Principal Bundle Structure of Matrix Manifolds. Mathematics 2021, 9, 1669. https://doi.org/10.3390/math9141669

Academic Editor: Marian Ioan Munteanu

Received: 11 June 2021
Accepted: 14 July 2021
Published: 15 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

<figure><img src="image_2.png" /></figure>

**Copyright:** © 2021 by the authors. License MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

1. Introduction

Low-rank matrices appear in many applications involving high-dimensional data. Low-rank models are commonly used in statistics, machine learning or data analysis (see [1] for a recent survey). Furthermore, low-rank approximation of matrices is the cornerstone of many modern numerical methods for high-dimensional problems in computational science, such as model-order-reduction methods for dynamical systems or parameter-dependent or stochastic equations [2–5].

These applications yield problems of approximation or optimization in the sets of matrices with fixed rank:

$$ M_r(\mathbb{R}^{n \times m}) = \{Z \in \mathbb{R}^{n \times m} : \mathrm{rank}(Z) = r\}. $$

Fixed-rank matrices appear also in the theory of characteristics of Partial Differential Equations and Monge-Ampère equations [6]. More precisely, it has been proven [6,7] that Monge-Ampère equations with $n$ independent variables and of Goursat-type are in one-to-one correspondence with the set $\{Z \in M_r(\mathbb{R}^{n \times n}) : r \le 2\}$. Thus, the parabolic or hyperbolic nature of the Monge-Ampère equation is related to the rank of such matrices.

In [8,9], the authors point out that Algebraic Geometry appears as a natural tool in study of the set $M_r(\mathbb{R}^{n \times m})$. We wish to mention the papers [10–12] that raise the natural question of how large these matrix spaces are.

<footnote n="1">¹ Department of Computer Science and Mathematics, Ecole Centrale de Nantes, 1 Rue de la Noë, BP 92101, CEDEX 3, 44321 Nantes, France; marie.billaud-friess@ec-nantes.fr (M.B.-F.); anthony.nouy@ec-nantes.fr (A.N.)</footnote>

<footnote n="2">² Departamento de Matemáticas, Física y Ciencias Tecnológicas, Universidad CEU Cardenal Herrera, CEU Universities, San Bartolomé 55, 46115 Alfara del Patriarca, Spain</footnote>

<footnote n="3">³ ESI International Chair@CEU-UCH, Universidad Cardenal Herrera-CEU, CEU Universities San Bartolomé 55, 46115 Alfara del Patriarca, Spain</footnote>

<footnote>* Correspondence: afalco@uchceu.es</footnote>
A usual geometric approach is to endow the set $M_r(\mathbb{R}^{n \times m})$ with the structure of a Riemannian manifold [13,14], which is seen as an embedded submanifold of $\mathbb{R}^{n \times m}$ equipped with the topology $\tau_{\mathbb{R}^{n \times m}}$ given by matrix norms. Standard algorithms then work in the ambient matrix space $\mathbb{R}^{n \times m}$ and do not rely on an explicit geometric description of the manifold using local charts (see, e.g., [15–18]). However, the matrix rank considered as a map is not continuous for the topology $\tau_{\mathbb{R}^{n \times m}}$, which can yield undesirable numerical issues.

The purpose of this paper is to propose a new geometric description of the sets of matrices with fixed rank, which is amenable for numerical use, and relies on the natural parametrization of matrices in $M_r(\mathbb{R}^{n \times m})$ given by

$$Z = UGV^T, \qquad (1)$$

where $U \in \mathbb{R}^{n \times r}$ and $V \in \mathbb{R}^{m \times r}$ are matrices with full rank $r < \min\{n, m\}$ and $G \in \mathbb{R}^{r \times r}$ is a non singular matrix. The set $M_r(\mathbb{R}^{n \times m})$ is here endowed with the structure of analytic principal bundle with an explicit description of local charts. This results in a description of the matrix space $\mathbb{R}^{n \times m}$ as an analytic manifold with a topology induced by local charts that is different from $\tau_{\mathbb{R}^{n \times m}}$ and for which the rank is a continuous map. Note that the representation (1) of a matrix Z is not unique because $Z = (UP)(P^{-1}GP^T)(VP^{-1})^T$ holds for every invertible matrix P in $\mathbb{R}^{r \times r}$. An argument used to dodge this undesirable property is the possibility to uniquely define a tangent space (see for example Section 2.1 in [18]), which is a prerequisite for standard algorithms on differentiable manifolds. The geometric description proposed in this paper avoids this undesirable property. Indeed, the system of local charts for the set $M_r(\mathbb{R}^{n \times m})$ is indexed on the set itself. This allows a natural definition of a neighbourhood for a matrix where all matrices admit a unique representation.

The present work opens the route for new numerical methods for optimization and dynamical low-rank approximation with algorithms working in local coordinates and avoiding the use of a Riemannian structure. In [19], such a framework is introduced for generalising iterative methods in optimization from Euclidean space to manifolds, which ensures that local convergence rates are preserved. Recently, a splitting algorithm relying on the geometric description of the set of fixed rank matrices proposed in this paper has been introduced for dynamical low-rank approximation [20].

The introduction of a principal bundle representation of matrix manifolds is also motivated by the importance of this geometric structure in the concept of gauge potential in physics [21].

Note that the proposed geometric description has a natural extension to the case of fixed-rank operators on infinite dimensional spaces and is consistent with the geometric description of manifolds of tensors with fixed rank proposed by Falcó, Hackbush and Nouy [22] in a tensor Banach space framework.

Before introducing the main results and outline of the paper, we recall some elements of geometry.

### 1.1. Elements of Geometry

In this paper, we follow the approach of Serge Lang [23] for the definition of a manifold $M$. In this framework, a set $M$ is equipped with an atlas which gives $M$ the structure of a topological space, with a topology induced by local charts, and the structure of differentiable manifold compatible with this topology. More precisely, the starting point is the definition of a collection of non-empty subsets $U_\alpha \subset M$, with $\alpha$ in a set $A$, such that $\{U_\alpha\}_{\alpha \in A}$ is a covering of $M$. The next step is the explicit construction for any $\alpha \in A$ of a local chart $\varphi_\alpha$ which is a bijection from $U_\alpha$ to an open set $X_\alpha$ of the finite dimensional space $\mathbb{R}^{N_\alpha}$ such that for any pair $\alpha, \alpha' \in M$ such that $U_\alpha \cap U_{\alpha'} \neq \emptyset$, the following properties hold:

(i) $\varphi_\alpha(U_\alpha \cap U_{\alpha'})$ and $\varphi_{\alpha'}(U_\alpha \cap U_{\alpha'})$ are open sets in $X_\alpha$ and $X_{\alpha'}$ respectively, and

(ii) the map

$$\varphi_{\alpha'} \circ \varphi_{\alpha}^{-1} : \varphi_{\alpha}(U_{\alpha} \cap U_{\alpha'}) \to \varphi_{\alpha'}(U_{\alpha} \cap U_{\alpha'})$$
A usual geometric approach is to endow the set $M_r(\mathbb{R}^{n \times m})$ with the structure of a Riemannian manifold [13,14], which is seen as an embedded submanifold of $\mathbb{R}^{n \times m}$ equipped with the topology $\tau_{\mathbb{R}^{n \times m}}$ given by matrix norms. Standard algorithms then work in the ambient matrix space $\mathbb{R}^{n \times m}$ and do not rely on an explicit geometric description of the manifold using local charts (see, e.g., [15–18]). However, the matrix rank considered as a map is not continuous for the topology $\tau_{\mathbb{R}^{n \times m}}$, which can yield undesirable numerical issues.

The purpose of this paper is to propose a new geometric description of the sets of matrices with fixed rank, which is amenable for numerical use, and relies on the natural parametrization of matrices in $M_r(\mathbb{R}^{n \times m})$ given by

$$Z = UGV^T, \qquad (1)$$

where $U \in \mathbb{R}^{n \times r}$ and $V \in \mathbb{R}^{m \times r}$ are matrices with full rank $r < \min\{n, m\}$ and $G \in \mathbb{R}^{r \times r}$ is a non singular matrix. The set $M_r(\mathbb{R}^{n \times m})$ is here endowed with the structure of analytic principal bundle with an explicit description of local charts. This results in a description of the matrix space $\mathbb{R}^{n \times m}$ as an analytic manifold with a topology induced by local charts that is different from $\tau_{\mathbb{R}^{n \times m}}$ and for which the rank is a continuous map. Note that the representation (1) of a matrix Z is not unique because $Z = (UP)(P^{-1}GP^T)(VP^{-1})^T$ holds for every invertible matrix P in $\mathbb{R}^{r \times r}$. An argument used to dodge this undesirable property is the possibility to uniquely define a tangent space (see for example Section 2.1 in [18]), which is a prerequisite for standard algorithms on differentiable manifolds. The geometric description proposed in this paper avoids this undesirable property. Indeed, the system of local charts for the set $M_r(\mathbb{R}^{n \times m})$ is indexed on the set itself. This allows a natural definition of a neighbourhood for a matrix where all matrices admit a unique representation.

The present work opens the route for new numerical methods for optimization and dynamical low-rank approximation with algorithms working in local coordinates and avoiding the use of a Riemannian structure. In [19], such a framework is introduced for generalising iterative methods in optimization from Euclidean space to manifolds, which ensures that local convergence rates are preserved. Recently, a splitting algorithm relying on the geometric description of the set of fixed rank matrices proposed in this paper has been introduced for dynamical low-rank approximation [20].

The introduction of a principal bundle representation of matrix manifolds is also motivated by the importance of this geometric structure in the concept of gauge potential in physics [21].

Note that the proposed geometric description has a natural extension to the case of fixed-rank operators on infinite dimensional spaces and is consistent with the geometric description of manifolds of tensors with fixed rank proposed by Falcó, Hackbush and Nouy [22] in a tensor Banach space framework.

Before introducing the main results and outline of the paper, we recall some elements of geometry.

### 1.1. Elements of Geometry

In this paper, we follow the approach of Serge Lang [23] for the definition of a manifold $M$. In this framework, a set $M$ is equipped with an atlas which gives $M$ the structure of a topological space, with a topology induced by local charts, and the structure of differentiable manifold compatible with this topology. More precisely, the starting point is the definition of a collection of non-empty subsets $U_\alpha \subset M$, with $\alpha$ in a set $A$, such that $\{U_\alpha\}_{\alpha \in A}$ is a covering of $M$. The next step is the explicit construction for any $\alpha \in A$ of a local chart $\varphi_\alpha$ which is a bijection from $U_\alpha$ to an open set $X_\alpha$ of the finite dimensional space $\mathbb{R}^{N_\alpha}$ such that for any pair $\alpha, \alpha' \in M$ such that $U_\alpha \cap U_{\alpha'} \neq \emptyset$, the following properties hold:

(i) $\varphi_\alpha(U_\alpha \cap U_{\alpha'})$ and $\varphi_{\alpha'}(U_\alpha \cap U_{\alpha'})$ are open sets in $X_\alpha$ and $X_{\alpha'}$ respectively, and

(ii) the map

$$\varphi_{\alpha'} \circ \varphi_{\alpha}^{-1} : \varphi_{\alpha}(U_{\alpha} \cap U_{\alpha'}) \to \varphi_{\alpha'}(U_{\alpha} \cap U_{\alpha'})$$
is a $C^p$ differentiable diffeomorphism, with $p \in \mathbb{N} \cup \{\infty\}$ or $p = \omega$ when the map is analytic.

Under the above assumptions, the set $\mathcal{A} := \{(U_\alpha, \varphi_\alpha) : \alpha \in A\}$ is an atlas which endows $\mathbb{M}$ with a structure of $C^p$ manifold. Then, we can say that $(\mathbb{M}, \mathcal{A})$ is a $C^p$ manifold, or an analytic manifold when $p = \omega$. A consequence of condition (ii) is that when $U_\alpha \cap U_{\alpha'} \neq \emptyset$ holds for $\alpha, \alpha' \in A$, then $N_\alpha = N_{\alpha'}$. In the particular case where $N_\alpha = N$ for all $\alpha \in A$, we say that $(\mathbb{M}, \mathcal{A})$ is a $C^p$ manifold modelled on $\mathbb{R}^N$. Otherwise, we say that it is a manifold not modelled on a particular finite-dimensional space. A paradigmatic example is the Grassmann manifold $\mathbb{G}(\mathbb{R}^k)$ of all linear subspaces of $\mathbb{R}^k$, such that

$$ \mathbb{G}(\mathbb{R}^k) = \bigcup_{0 \le r \le k} \mathbb{G}_r(\mathbb{R}^k), $$

where $\mathbb{G}_0(\mathbb{R}^k) = \{0\}$ and $\mathbb{G}_k(\mathbb{R}^k) = \{\mathbb{R}^k\}$ are trivial manifolds and $\mathbb{G}_r(\mathbb{R}^k)$ is a manifold modelled on the linear space $\mathbb{R}^{(k-r) \times r}$ for $0 < r < k$. Consequently, $\mathbb{G}(\mathbb{R}^k)$ is a manifold not modelled on a particular finite-dimensional space.

The atlas also endows $\mathbb{M}$ with a topology given by

$$ \tau_{\mathcal{A}} := \{ \varphi_{\alpha}^{-1}(O) : \alpha \in A \text{ and } O \text{ an open set in } X_{\alpha} \}, $$

which makes $(\mathbb{M}, \tau_{\mathcal{A}})$ a topological space where each local chart

$$ \varphi_{\alpha} : (U_{\alpha}, \tau_{\mathcal{A}}|_{U_{\alpha}}) \to (X_{\alpha}, \tau_{\mathbb{R}^{n_{\alpha}}}|_{X_{\alpha}}), $$

considered as a map between topological spaces is a homeomorphism. (Here $(\mathcal{X}, \tau)$ denotes a topological space, and if $\mathcal{X}' \subset \mathcal{X}$, then $\tau|_{\mathcal{X}'}$ denotes the subspace topology.)

## 1.2. Main Results and Outline

Our first remark is that the matrix space $\mathbb{R}^{n \times m}$ is an analytic manifold modelled on itself, and its geometric structure is fully compatible with the topology $\tau_{\mathbb{R}^{n \times m}}$ induced by a matrix norm. In this paper, we define an atlas on $\mathcal{M}_r(\mathbb{R}^{n \times m})$, which gives this set the structure of an analytic manifold, with a topology induced by the atlas fully compatible with the subspace topology $\tau_{\mathbb{R}^{n \times m}|_{\mathcal{M}_r(\mathbb{R}^{n \times m})}}$. This implies that $\mathcal{M}_r(\mathbb{R}^{n \times m})$ is an embedded submanifold of the matrix manifold $\mathbb{R}^{n \times m}$ modelled on itself. (Note that the set $\mathcal{M}_0(\mathbb{R}^{n \times m}) = \{0\}$ is a trivial manifold, which is trivially embedded in $\mathbb{R}^{n \times m}$.) For the topology $\tau_{\mathbb{R}^{n \times m}}$, the matrix rank considered as a map is not continuous but only lower semi-continuous. However, if $\mathbb{R}^{n \times m}$ is seen as the disjoint union of sets of matrices with fixed rank,

$$ \mathbb{R}^{n \times m} = \bigcup_{0 \le r \le \min\{n, m\}} \mathcal{M}_r(\mathbb{R}^{n \times m}), \quad (2) $$

then $\mathbb{R}^{n \times m}$ has the structure of an analytic manifold not modelled on a particular finite-dimensional space equipped with a topology

$$ \tau_{\mathbb{R}^{n \times m}}^* = \bigcup_{0 \le r \le \min\{n, m\}} \tau_{\mathbb{R}^{n \times m}|_{\mathcal{M}_r(\mathbb{R}^{n \times m})}}, $$

which is not equivalent to $\tau_{\mathbb{R}^{n \times m}}$, and for which the matrix rank is a continuous map.

Note that in the case where $r = n = m$, the set $\mathcal{M}_n(\mathbb{R}^{n \times n})$ coincides with the general linear group $\mathrm{GL}_n$ of invertible matrices in $\mathbb{R}^{n \times n}$, which is an analytic manifold trivially embedded in $\mathbb{R}^{n \times n}$. In all other cases are addressed in this paper, our geometric description of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ relies on a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$, with $k = n$ or $m$.

Therefore, we start in Section 2 by introducing a geometric description of $\mathbb{G}_r(\mathbb{R}^k)$. A classical approach consists of describing $\mathbb{G}_r(\mathbb{R}^k)$ as the quotient manifold $\mathcal{M}_r(\mathbb{R}^{k \times r})/\mathrm{GL}_r$
is a $C^p$ differentiable diffeomorphism, with $p \in \mathbb{N} \cup \{\infty\}$ or $p = \omega$ when the map is analytic.

Under the above assumptions, the set $\mathcal{A} := \{(U_\alpha, \varphi_\alpha) : \alpha \in A\}$ is an atlas which endows $\mathbb{M}$ with a structure of $C^p$ manifold. Then, we can say that $(\mathbb{M}, \mathcal{A})$ is a $C^p$ manifold, or an analytic manifold when $p = \omega$. A consequence of condition (ii) is that when $U_\alpha \cap U_{\alpha'} \neq \emptyset$ holds for $\alpha, \alpha' \in A$, then $N_\alpha = N_{\alpha'}$. In the particular case where $N_\alpha = N$ for all $\alpha \in A$, we say that $(\mathbb{M}, \mathcal{A})$ is a $C^p$ manifold modelled on $\mathbb{R}^N$. Otherwise, we say that it is a manifold not modelled on a particular finite-dimensional space. A paradigmatic example is the Grassmann manifold $\mathbb{G}(\mathbb{R}^k)$ of all linear subspaces of $\mathbb{R}^k$, such that

$$ \mathbb{G}(\mathbb{R}^k) = \bigcup_{0 \le r \le k} \mathbb{G}_r(\mathbb{R}^k), $$

where $\mathbb{G}_0(\mathbb{R}^k) = \{0\}$ and $\mathbb{G}_k(\mathbb{R}^k) = \{\mathbb{R}^k\}$ are trivial manifolds and $\mathbb{G}_r(\mathbb{R}^k)$ is a manifold modelled on the linear space $\mathbb{R}^{(k-r) \times r}$ for $0 < r < k$. Consequently, $\mathbb{G}(\mathbb{R}^k)$ is a manifold not modelled on a particular finite-dimensional space.

The atlas also endows $\mathbb{M}$ with a topology given by

$$ \tau_{\mathcal{A}} := \{ \varphi_{\alpha}^{-1}(O) : \alpha \in A \text{ and } O \text{ an open set in } X_{\alpha} \}, $$

which makes $(\mathbb{M}, \tau_{\mathcal{A}})$ a topological space where each local chart

$$ \varphi_{\alpha} : (U_{\alpha}, \tau_{\mathcal{A}}|_{U_{\alpha}}) \to (X_{\alpha}, \tau_{\mathbb{R}^{n_{\alpha}}}|_{X_{\alpha}}), $$

considered as a map between topological spaces is a homeomorphism. (Here $(\mathcal{X}, \tau)$ denotes a topological space, and if $\mathcal{X}' \subset \mathcal{X}$, then $\tau|_{\mathcal{X}'}$ denotes the subspace topology.)

## 1.2. Main Results and Outline

Our first remark is that the matrix space $\mathbb{R}^{n \times m}$ is an analytic manifold modelled on itself, and its geometric structure is fully compatible with the topology $\tau_{\mathbb{R}^{n \times m}}$ induced by a matrix norm. In this paper, we define an atlas on $\mathcal{M}_r(\mathbb{R}^{n \times m})$, which gives this set the structure of an analytic manifold, with a topology induced by the atlas fully compatible with the subspace topology $\tau_{\mathbb{R}^{n \times m}|_{\mathcal{M}_r(\mathbb{R}^{n \times m})}}$. This implies that $\mathcal{M}_r(\mathbb{R}^{n \times m})$ is an embedded submanifold of the matrix manifold $\mathbb{R}^{n \times m}$ modelled on itself. (Note that the set $\mathcal{M}_0(\mathbb{R}^{n \times m}) = \{0\}$ is a trivial manifold, which is trivially embedded in $\mathbb{R}^{n \times m}$.) For the topology $\tau_{\mathbb{R}^{n \times m}}$, the matrix rank considered as a map is not continuous but only lower semi-continuous. However, if $\mathbb{R}^{n \times m}$ is seen as the disjoint union of sets of matrices with fixed rank,

$$ \mathbb{R}^{n \times m} = \bigcup_{0 \le r \le \min\{n,m\}} \mathcal{M}_r(\mathbb{R}^{n \times m}), \quad (2) $$

then $\mathbb{R}^{n \times m}$ has the structure of an analytic manifold not modelled on a particular finite-dimensional space equipped with a topology

$$ \tau_{\mathbb{R}^{n \times m}}^* = \bigcup_{0 \le r \le \min\{n,m\}} \tau_{\mathbb{R}^{n \times m}|_{\mathcal{M}_r(\mathbb{R}^{n \times m})}}, $$

which is not equivalent to $\tau_{\mathbb{R}^{n \times m}}$, and for which the matrix rank is a continuous map.

Note that in the case where $r = n = m$, the set $\mathcal{M}_n(\mathbb{R}^{n \times n})$ coincides with the general linear group $\mathrm{GL}_n$ of invertible matrices in $\mathbb{R}^{n \times n}$, which is an analytic manifold trivially embedded in $\mathbb{R}^{n \times n}$. In all other cases are addressed in this paper, our geometric description of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ relies on a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$, with $k = n$ or $m$.

Therefore, we start in Section 2 by introducing a geometric description of $\mathbb{G}_r(\mathbb{R}^k)$. A classical approach consists of describing $\mathbb{G}_r(\mathbb{R}^k)$ as the quotient manifold $\mathcal{M}_r(\mathbb{R}^{k \times r})/\mathrm{GL}_r$
of equivalent classes of full-rank matrices Z in $M_r(\mathbb{R}^{k \times r})$ with the same column space $\text{col}_{k,r}(Z)$. Here, we avoid the use of equivalent classes and provide an explicit description of an atlas $\mathcal{A}_{k,r} = \{(\mathfrak{U}_Z, \varphi_Z)\}_{Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})}$ for $\mathbb{G}_r(\mathbb{R}^k)$, with local chart

$$ \varphi_Z : \mathfrak{U}_Z \to \mathbb{R}^{(k-r) \times r}, \quad \varphi_Z^{-1}(X) = \text{col}_{k,r}(Z + Z_\perp X), $$

where $Z_\perp \in \mathbb{R}^{k \times (k-r)}$ is such that $Z_\perp^T Z = 0$ (see Remark 1 for a practical choice) and $\text{col}_{k,r}(A)$ denotes the column space of a matrix $A \in \mathbb{R}^{k \times r}$, and we prove that the neighbourhood $\mathfrak{U}_Z$ has the structure of a Lie group. This parametrization of the Grassmann manifold is introduced in ([24] Section 2), but the authors do not elaborate on it.

Then, in Section 3, we consider the particular case of full-rank matrices. We introduce an atlas $\mathcal{B}_{k,r} = \{(\mathcal{V}_Z, \xi_Z)\}_{Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})}$ for the manifold $M_r(\mathbb{R}^{k \times r})$ of matrices with full rank $r < k$, with local chart

$$ \xi_Z : \mathcal{V}_Z \to \mathbb{R}^{(k-r) \times r} \times \text{GL}_r, \quad \xi_Z^{-1}(X, G) = (Z + Z_\perp X)G, $$

and prove that $M_r(\mathbb{R}^{k \times r})$ is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\text{GL}_r$. Moreover, we prove that $M_r(\mathbb{R}^{k \times r})$ is an embedded submanifold of $(\mathbb{R}^{k \times r}, \tau_{\mathbb{R}^{k \times r}}^*)$ and that each of the neighbourhoods $\mathcal{V}_Z$ have the structure of a Lie group.

Finally, in Section 4, we provide an analytic atlas $\mathcal{B}_{n,m,r} = \{(\mathcal{U}_Z, \theta_Z)\}_{Z \in \mathcal{M}_r(\mathbb{R}^{n \times m})}$ for the set $M_r(\mathbb{R}^{n \times m})$ of matrices $Z = UGV^T$ with rank $r < \min\{n, m\}$, with local chart

$$ \theta_Z : \mathcal{U}_Z \to \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r, \quad \theta_Z^{-1}(X, Y, H) = (U + U_\perp X)H(V + V_\perp Y), $$

and we prove that $M_r(\mathbb{R}^{n \times m})$ is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\text{GL}_r$. Then, we prove that $M_r(\mathbb{R}^{n \times m})$ is an embedded submanifold of $(\mathbb{R}^{n \times m}, \tau_{\mathbb{R}^{n \times m}}^*)$ and that each of the neighbourhoods $\mathcal{U}_Z$ have the structure of a Lie group.

## 2. The Grassmann Manifold $\mathbb{G}_r(\mathbb{R}^k)$

In this section, we present a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of all subspaces of dimension $r$ in $\mathbb{R}^k$, $0 < r < k$,

$$ \mathbb{G}_r(\mathbb{R}^k) = \{\mathcal{V} \subset \mathbb{R}^k : \mathcal{V} \text{ is a linear subspace with } \dim(\mathcal{V}) = r\}, $$

with an explicit description of local charts. We first introduce the surjective map

$$ \text{col}_{k,r} : M_r(\mathbb{R}^{k \times r}) \to \mathbb{G}_r(\mathbb{R}^k), \quad Z \mapsto \text{col}_{k,r}(Z), $$

where $\text{col}_{k,r}(Z)$ is the column space of the matrix $Z$, which is the subspace spanned by the column vectors of $Z$. Given $\mathcal{V} \in \mathbb{G}_r(\mathbb{R}^k)$, there are infinitely many matrices $Z$ such that $\text{col}_{k,r}(Z) = \mathcal{V}$. Given a matrix $Z \in M_r(\mathbb{R}^{k \times r})$, the set of matrices in $M_r(\mathbb{R}^{k \times r})$ with the same column space as $Z$ is

$$ ZGL_r := \{ZG : G \in \text{GL}_r\}. $$

### 2.1. An Atlas for $\mathbb{G}_r(\mathbb{R}^k)$

For a given matrix $Z$ in $M_r(\mathbb{R}^{k \times r})$, we let $Z_\perp \in \mathcal{M}_{k-r}(\mathbb{R}^{k \times (k-r)})$ be a matrix such that $Z^T Z_\perp = 0$, and we introduce an *affine cross section*

$$ S_Z := \{W \in M_r(\mathbb{R}^{k \times r}) : Z^T W = Z^T Z\}, \quad (3) $$

which has the following equivalent characterization.

**Lemma 1.** The affine cross section $S_Z$ is characterized by

$$ S_Z = \{Z + Z_\perp X : X \in \mathbb{R}^{(k-r) \times r}\}, \quad (4) $$
of equivalent classes of full-rank matrices Z in $M_r(\mathbb{R}^{k \times r})$ with the same column space $col_{k,r}(Z)$. Here, we avoid the use of equivalent classes and provide an explicit description of an atlas $A_{k,r} = \{(\mathfrak{U}_Z, \varphi_Z)\}_{Z \in M_r(\mathbb{R}^{k \times r})}$ for $\mathbb{G}_r(\mathbb{R}^k)$, with local chart

$$ \varphi_Z : \mathfrak{U}_Z \to \mathbb{R}^{(k-r) \times r}, \quad \varphi_Z^{-1}(X) = \mathrm{col}_{k,r}(Z + Z_\perp X), $$

where $Z_\perp \in \mathbb{R}^{k \times (k-r)}$ is such that $Z_\perp^T Z = 0$ (see Remark 1 for a practical choice) and $\mathrm{col}_{k,r}(A)$ denotes the column space of a matrix $A \in \mathbb{R}^{k \times r}$, and we prove that the neighbourhood $\mathfrak{U}_Z$ has the structure of a Lie group. This parametrization of the Grassmann manifold is introduced in ([24] Section 2), but the authors do not elaborate on it.

Then, in Section 3, we consider the particular case of full-rank matrices. We introduce an atlas $B_{k,r} = \{(\mathcal{V}_Z, \xi_Z)\}_{Z \in M_r(\mathbb{R}^{k \times r})}$ for the manifold $M_r(\mathbb{R}^{k \times r})$ of matrices with full rank $r < k$, with local chart

$$ \xi_Z : \mathcal{V}_Z \to \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r, \quad \xi_Z^{-1}(X, G) = (Z + Z_\perp X)G, $$

and prove that $M_r(\mathbb{R}^{k \times r})$ is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$. Moreover, we prove that $M_r(\mathbb{R}^{k \times r})$ is an embedded submanifold of $(\mathbb{R}^{k \times r}, \tau_{\mathbb{R}^{k \times r}}^*)$ and that each of the neighbourhoods $\mathcal{V}_Z$ have the structure of a Lie group.

Finally, in Section 4, we provide an analytic atlas $B_{n,m,r} = \{(\mathcal{U}_Z, \theta_Z)\}_{Z \in M_r(\mathbb{R}^{n \times m})}$ for the set $M_r(\mathbb{R}^{n \times m})$ of matrices $Z = UGV^T$ with rank $r < \min\{n, m\}$, with local chart

$$ \theta_Z : \mathcal{U}_Z \to \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r, \quad \theta_Z^{-1}(X, Y, H) = (U + U_\perp X)H(V + V_\perp Y), $$

and we prove that $M_r(\mathbb{R}^{n \times m})$ is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$. Then, we prove that $M_r(\mathbb{R}^{n \times m})$ is an embedded submanifold of $(\mathbb{R}^{n \times m}, \tau_{\mathbb{R}^{n \times m}}^*)$ and that each of the neighbourhoods $\mathcal{U}_Z$ have the structure of a Lie group.

## 2. The Grassmann Manifold $\mathbb{G}_r(\mathbb{R}^k)$

In this section, we present a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of all subspaces of dimension $r$ in $\mathbb{R}^k$, $0 < r < k$,

$$ \mathbb{G}_r(\mathbb{R}^k) = \{\mathcal{V} \subset \mathbb{R}^k : \mathcal{V} \text{ is a linear subspace with } \dim(\mathcal{V}) = r\}, $$

with an explicit description of local charts. We first introduce the surjective map

$$ \mathrm{col}_{k,r} : M_r(\mathbb{R}^{k \times r}) \to \mathbb{G}_r(\mathbb{R}^k), \quad Z \mapsto \mathrm{col}_{k,r}(Z), $$

where $\mathrm{col}_{k,r}(Z)$ is the column space of the matrix $Z$, which is the subspace spanned by the column vectors of $Z$. Given $\mathcal{V} \in \mathbb{G}_r(\mathbb{R}^k)$, there are infinitely many matrices $Z$ such that $\mathrm{col}_{k,r}(Z) = \mathcal{V}$. Given a matrix $Z \in M_r(\mathbb{R}^{k \times r})$, the set of matrices in $M_r(\mathbb{R}^{k \times r})$ with the same column space as $Z$ is

$$ ZGL_r := \{ZG : G \in GL_r\}. $$

### 2.1. An Atlas for $\mathbb{G}_r(\mathbb{R}^k)$

For a given matrix $Z$ in $M_r(\mathbb{R}^{k \times r})$, we let $Z_\perp \in M_{k-r}(\mathbb{R}^{k \times (k-r)})$ be a matrix such that $Z^T Z_\perp = 0$, and we introduce an *affine cross section*

$$ S_Z := \{W \in M_r(\mathbb{R}^{k \times r}) : Z^T W = Z^T Z\}, \quad (3) $$

which has the following equivalent characterization.

**Lemma 1.** *The affine cross section $S_Z$ is characterized by*

$$ S_Z = \{Z + Z_\perp X : X \in \mathbb{R}^{(k-r) \times r}\}, \quad (4) $$
and the map

$$ \eta_Z : \mathbb{R}^{(k-r) \times r} \to S_Z, \quad X \mapsto Z + Z_{\perp} X $$

is bijective.

**Proof.** We first observe that $Z^T(Z + Z_{\perp}X_Z) = Z^T Z$ for all $X \in \mathbb{R}^{(k-r) \times r}$, which implies that $\{Z + Z_{\perp}X : X \in \mathbb{R}^{(k-r) \times r}\} \subset S_Z$. For the other inclusion, we observe that if $W \in S_Z$, then $Z^T W = Z^T Z$ and hence $W - Z \in \text{col}_{k,r}(Z)^{\perp}$, the orthogonal subspace to $\text{col}_{k,r}(Z)$ in $\mathbb{R}^k$. Since $\text{col}_{k,r}(Z)^{\perp} = \text{col}_{k,k-r}(Z_{\perp})$, there exists $X \in \mathbb{R}^{(k-r) \times r}$ such that $W - Z = Z_{\perp}X$. Proving that $\eta_Z$ is bijective is straightforward. $\square$

**Proposition 1.** For each $W \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ such that $\det(Z^T W) \neq 0$, there exists a unique $G_W \in \text{GL}_r$ such that

$$ W\text{GL}_r \cap S_Z = \{WG_W^{-1}\} $$

holds, which means that the set of matrices with the same column space as $W$ intersects $S_Z$ at the single point $WG_W^{-1}$. Furthermore, $G_W = \text{id}_r$ if and only if $W \in S_Z$.

**Proof.** By Lemma 1, a matrix $A \in W\text{GL}_r \cap S_Z$ is such that $A = WG_W^{-1} = Z + Z_{\perp}X$ for a certain $G_W \in \text{GL}_r$ and a certain $X \in \mathbb{R}^{(k-r) \times r}$. Then $Z^T WG_W^{-1} = Z^T Z$ and $G_W$ is uniquely defined by $G_W = (Z^T Z)^{-1}(Z^T W)$, which proves that $W\text{GL}_r \cap S_Z$ is the singleton $\{WG_W^{-1}\}$, and $G_W = \text{id}_r$ if and only if $W \in S_Z$. $\square$

**Corollary 1.** For each $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, the map $\text{col}_{k,r} : S_Z \to \mathbb{G}_r(\mathbb{R}^k)$ is injective.

**Proof.** Let us assume the existence of $W, \tilde{W} \in S_Z$ such that $\text{col}_{k,r}(W) = \text{col}_{k,r}(\tilde{W})$. Then $W = \tilde{W}$ by Proposition 1. $\square$

Lemma 1 and Corollary 1 allow us to construct a system of local charts for $\mathbb{G}_r(\mathbb{R}^k)$ by defining for each $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ a neighbourhood of $\text{col}_{k,r}(Z)$ by

$$ \mathfrak{U}_Z := \text{col}_{k,r}(S_Z) = \{\text{col}_{k,r}(W) : W \in S_Z\} $$

together with the bijective map

$$ \varphi_Z := (\text{col}_{k,r} \circ \eta_Z)^{-1} : \mathfrak{U}_Z \to \mathbb{R}^{(k-r) \times r} $$

such that

$$ \varphi_Z^{-1}(X) = \text{col}_{k,r}(Z + Z_{\perp}X) $$

for $X \in \mathbb{R}^{(k-r) \times r}$. We denote by $Z^+$ the Moore-Penrose pseudo-inverse of the full rank matrix $Z \in \mathcal{M}_r(\mathbb{R}^{r \times k})$, defined by

$$ Z^+ := (Z^T Z)^{-1} Z^T \in \mathcal{M}_r(\mathbb{R}^{r \times k}). $$

It satisfies $Z^+Z = \text{id}_r$ and $Z^+Z_{\perp} = 0$. Moreover, $ZZ^+ \in \mathbb{R}^{k \times k}$ is the projection onto $\text{col}_{k,r}(Z)$ parallel to $\text{col}_{k,r}(Z)^{\perp}$. Finally, we have the following result.

**Theorem 1.** The collection $\mathcal{A}_{k,r} := \{(\mathfrak{U}_Z, \varphi_Z) : Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})\}$ is an analytic atlas for $\mathbb{G}_r(\mathbb{R}^k)$ and hence $(\mathbb{G}_r(\mathbb{R}^k), \mathcal{A}_{k,r})$ is an analytic $r(k-r)$-dimensional manifold modelled on $\mathbb{R}^{(k-r) \times r}$.

**Proof.** Clearly $\{\mathfrak{U}_Z\}_{Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})}$ is a covering of $\mathbb{G}_r(\mathbb{R}^k)$. Now let $Z$ and $\tilde{Z}$ be such that $\mathfrak{U}_Z \cap \mathfrak{U}_Z \neq \emptyset$. Let $\mathcal{V} \in \mathfrak{U}_Z$ such that $\mathcal{V} = \varphi_Z^{-1}(X) = \text{col}_{k,r}(Z + Z_{\perp}X)$, with $X \in \mathbb{R}^{k \times (k-r)}$. We can write $Z + Z_{\perp}X = (\tilde{Z} + \tilde{Z}_{\perp} \tilde{X})G$ with $G = \tilde{Z}^+(Z + Z_{\perp}X)$ and $\tilde{X} = \tilde{Z}_{\perp}^+(Z + Z_{\perp}X)G^{-1}$. Therefore, $\mathcal{V} = \text{col}_{k,r}((\tilde{Z} + \tilde{Z}_{\perp} \tilde{X})G) = \text{col}_{k,r}(\tilde{Z} + \tilde{Z}_{\perp} \tilde{X}) = \varphi_{\tilde{Z}}^{-1}(\tilde{X}) \in \mathfrak{U}_{\tilde{Z}}$, which implies that $\mathfrak{U}_Z = \mathfrak{U}_Z \cap \mathfrak{U}_{\tilde{Z}}$. Therefore, $\varphi_Z(\mathfrak{U}_Z \cap \mathfrak{U}_{\tilde{Z}}) = \varphi_Z(\mathfrak{U}_Z) = \mathbb{R}^{k \times (n-k)}$ is an open set. In the
and the map

$$ \eta_Z : \mathbb{R}^{(k-r) \times r} \to S_Z, \quad X \mapsto Z + Z_{\perp} X $$

is bijective.

**Proof.** We first observe that $Z^T(Z + Z_{\perp}X_Z) = Z^T Z$ for all $X \in \mathbb{R}^{(k-r) \times r}$, which implies that $\{Z + Z_{\perp}X : X \in \mathbb{R}^{(k-r) \times r}\} \subset S_Z$. For the other inclusion, we observe that if $W \in S_Z$, then $Z^T W = Z^T Z$ and hence $W - Z \in \text{col}_{k,r}(Z)^{\perp}$, the orthogonal subspace to $\text{col}_{k,r}(Z)$ in $\mathbb{R}^k$. Since $\text{col}_{k,r}(Z)^{\perp} = \text{col}_{k,k-r}(Z_{\perp})$, there exists $X \in \mathbb{R}^{(k-r) \times r}$ such that $W - Z = Z_{\perp}X$. Proving that $\eta_Z$ is bijective is straightforward. $\square$

**Proposition 1.** For each $W \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ such that $\det(Z^T W) \neq 0$, there exists a unique $G_W \in \text{GL}_r$ such that

$$ W\text{GL}_r \cap S_Z = \{WG_W^{-1}\} $$

holds, which means that the set of matrices with the same column space as $W$ intersects $S_Z$ at the single point $WG_W^{-1}$. Furthermore, $G_W = \text{id}_r$ if and only if $W \in S_Z$.

**Proof.** By Lemma 1, a matrix $A \in W\text{GL}_r \cap S_Z$ is such that $A = WG_W^{-1} = Z + Z_{\perp}X$ for a certain $G_W \in \text{GL}_r$ and a certain $X \in \mathbb{R}^{(k-r) \times r}$. Then $Z^T WG_W^{-1} = Z^T Z$ and $G_W$ is uniquely defined by $G_W = (Z^T Z)^{-1}(Z^T W)$, which proves that $W\text{GL}_r \cap S_Z$ is the singleton $\{WG_W^{-1}\}$, and $G_W = \text{id}_r$ if and only if $W \in S_Z$. $\square$

**Corollary 1.** For each $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, the map $\text{col}_{k,r} : S_Z \to \mathbb{G}_r(\mathbb{R}^k)$ is injective.

**Proof.** Let us assume the existence of $W, \tilde{W} \in S_Z$ such that $\text{col}_{k,r}(W) = \text{col}_{k,r}(\tilde{W})$. Then $W = \tilde{W}$ by Proposition 1. $\square$

Lemma 1 and Corollary 1 allow us to construct a system of local charts for $\mathbb{G}_r(\mathbb{R}^k)$ by defining for each $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ a neighbourhood of $\text{col}_{k,r}(Z)$ by

$$ \mathfrak{U}_Z := \text{col}_{k,r}(S_Z) = \{\text{col}_{k,r}(W) : W \in S_Z\} $$

together with the bijective map

$$ \varphi_Z := (\text{col}_{k,r} \circ \eta_Z)^{-1} : \mathfrak{U}_Z \to \mathbb{R}^{(k-r) \times r} $$

such that

$$ \varphi_Z^{-1}(X) = \text{col}_{k,r}(Z + Z_{\perp}X) $$

for $X \in \mathbb{R}^{(k-r) \times r}$. We denote by $Z^+$ the Moore-Penrose pseudo-inverse of the full rank matrix $Z \in \mathcal{M}_r(\mathbb{R}^{r \times k})$, defined by

$$ Z^+ := (Z^T Z)^{-1} Z^T \in \mathcal{M}_r(\mathbb{R}^{r \times k}). $$

It satisfies $Z^+Z = \text{id}_r$ and $Z^+Z_{\perp} = 0$. Moreover, $ZZ^+ \in \mathbb{R}^{k \times k}$ is the projection onto $\text{col}_{k,r}(Z)$ parallel to $\text{col}_{k,r}(Z)^{\perp}$. Finally, we have the following result.

**Theorem 1.** The collection $\mathcal{A}_{k,r} := \{(\mathfrak{U}_Z, \varphi_Z) : Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})\}$ is an analytic atlas for $\mathbb{G}_r(\mathbb{R}^k)$ and hence $(\mathbb{G}_r(\mathbb{R}^k), \mathcal{A}_{k,r})$ is an analytic $r(k-r)$-dimensional manifold modelled on $\mathbb{R}^{(k-r) \times r}$.

**Proof.** Clearly $\{\mathfrak{U}_Z\}_{Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})}$ is a covering of $\mathbb{G}_r(\mathbb{R}^k)$. Now let $Z$ and $\tilde{Z}$ be such that $\mathfrak{U}_Z \cap \mathfrak{U}_Z \neq \emptyset$. Let $\mathcal{V} \in \mathfrak{U}_Z$ such that $\mathcal{V} = \varphi_Z^{-1}(X) = \text{col}_{k,r}(Z + Z_{\perp}X)$, with $X \in \mathbb{R}^{k \times (k-r)}$. We can write $Z + Z_{\perp}X = (\tilde{Z} + \tilde{Z}_{\perp} \tilde{X})G$ with $G = \tilde{Z}^+(Z + Z_{\perp}X)$ and $\tilde{X} = \tilde{Z}_{\perp}^+(Z + Z_{\perp}X)G^{-1}$. Therefore, $\mathcal{V} = \text{col}_{k,r}((\tilde{Z} + \tilde{Z}_{\perp} \tilde{X})G) = \text{col}_{k,r}(\tilde{Z} + \tilde{Z}_{\perp} \tilde{X}) = \varphi_{\tilde{Z}}^{-1}(\tilde{X}) \in \mathfrak{U}_{\tilde{Z}}$, which implies that $\mathfrak{U}_Z = \mathfrak{U}_Z \cap \mathfrak{U}_{\tilde{Z}}$. Therefore, $\varphi_Z(\mathfrak{U}_Z \cap \mathfrak{U}_{\tilde{Z}}) = \varphi_Z(\mathfrak{U}_Z) = \mathbb{R}^{k \times (n-k)}$ is an open set. In the
same way, we show that $\mathfrak{U}_{\bar{Z}} = \mathfrak{U}_Z \cap \mathfrak{U}_{\bar{Z}}$ and $\varphi_{\bar{Z}}(\mathfrak{U}_Z) = \mathbb{R}^{k \times (n-k)}$ is an open set. Finally, the map $\varphi_{\bar{Z}} \circ \varphi_{\bar{Z}}^{-1}$ from $\mathbb{R}^{(k-r) \times r}$ to $\mathbb{R}^{(k-r) \times r}$ is given by $\varphi_{\bar{Z}} \circ \varphi_{\bar{Z}}^{-1}(X) = \tilde{Z}_{\perp}^{+}(Z + Z_{\perp}X)G^{-1}$, with $G = \tilde{Z}^{+}(Z + Z_{\perp}X_Z)$, which is clearly an analytic map. $\square$

**Remark 1.** A possible choice for $Z_{\perp}$ satisfying $Z_{\perp}^T Z = 0$ is $Z_{\perp} = (id_k - ZZ^+)B_{\perp}$ where $B_{\perp} \in \mathcal{M}_{k-r}(\mathbb{R}^{k \times (k-r)})$ is such that its column space is a complement of the column space of $Z$. In practice, we can determine a set of $r$ linear independent rows of $Z$ (see, e.g., [25,26]), with indices $I$, and then choose $B_{\perp}$ such that $(B_{\perp})_{i,j} = \delta_{i,j}$ if $i \notin I$ and $0$ if $i \in I$, for $1 \le i \le k, 1 \le j \le k-r$. For a given $X \in \mathbb{R}^{(k-r) \times r}$, the computation of $Z_{\perp}X$ does not require $Z_{\perp}$ and has a complexity $O(r^2k)$.

## 2.2. Lie Group Structure of Neighbourhoods $\mathfrak{U}_Z$

Here we prove that each neighbourhood $\mathfrak{U}_Z$ of $\mathfrak{G}_r(\mathbb{R}^k)$ is a Lie group. For that, we first note that a neighbourhood $\mathfrak{U}_Z$ of $\mathfrak{G}_r(\mathbb{R}^k)$ can be identified with the set $S_Z$ through the application $\mathrm{col}_{k,r}: S_Z \to \mathfrak{U}_Z$. The next step is to identify $S_Z$ with a closed Lie subgroup of $\mathrm{GL}_k$, denoted by $\mathcal{G}_Z$, with associated Lie algebra $\mathcal{g}_Z$ isomorphic to $\mathbb{R}^{r \times (k-r)}$, and such that the exponential map $\exp: \mathcal{g}_Z \to \mathcal{G}_Z$ is a diffeomorphism. (We recall that the matrix exponential $\exp: \mathbb{R}^{k \times k} \to \mathrm{GL}_k$ is defined by $\exp(A) = \sum_{n=0}^{\infty} \frac{A^n}{n!}$.) To this end, for a given $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, we introduce the vector space

$$ \mathfrak{g}_Z := \{Z_{\perp}XZ^+ : X \in \mathbb{R}^{(k-r) \times r}\} \subset \mathbb{R}^{k \times k}. \qquad (5) $$

The following proposition proves that $\mathfrak{g}_Z$ is a commutative subalgebra of $\mathbb{R}^{k \times k}$.

**Proposition 2.** For all $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$,

$$ (Z_{\perp}XZ^{+})(Z_{\perp}\tilde{X}Z^{+}) = 0 $$

holds, and $\mathfrak{g}_Z$ is a commutative subalgebra of $\mathbb{R}^{k \times k}$. Moreover,

$$ \exp(Z_{\perp}XZ^{+}) = id_{k} + Z_{\perp}XZ^{+}, \qquad (6) $$

$$ \exp(Z_{\perp}XZ^{+})Z = Z + Z_{\perp}X, \qquad (7) $$

and

$$ \exp(Z_{\perp}XZ^{+})Z_{\perp} = Z_{\perp} \qquad (8) $$

hold for all $X \in \mathbb{R}^{(k-r) \times r}$.

**Proof.** Since $(Z_{\perp}XZ^{+})(Z_{\perp}\tilde{X}Z^{+}) = 0$ holds for all $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$, the vector space $\mathfrak{g}_Z$ is a closed subalgebra of the matrix unitary algebra $\mathbb{R}^{k \times k}$. As a consequence, $(Z_{\perp}XZ^{+})^p = 0$ holds for all $X \in \mathbb{R}^{(k-r) \times r}$ and all $p \ge 2$, which proves (6). We directly deduce (7) using $ZZ^{+} = id_r$ and (8) using $Z^{+}Z_{\perp} = 0$. $\square$

From Proposition 2 and the definition of $S_Z$, we obtain the following results.

**Corollary 2.** The affine cross section $S_Z$ satisfies

$$ S_Z = \{\exp(Z_{\perp}XZ^{+})Z : X \in \mathbb{R}^{(k-r) \times r}\}, \qquad (9) $$

and

$$ [\exp(Z_{\perp}XZ^{+})Z | Z_{\perp}] \in \mathrm{GL}_{k} \qquad (10) $$

for all $X \in \mathbb{R}^{(k-r) \times r}$, where the brackets $[\cdot|\cdot]$ are used for matrix concatenation.
same way, we show that $\mathfrak{U}_{\bar{Z}} = \mathfrak{U}_Z \cap \mathfrak{U}_{\bar{Z}}$ and $\varphi_{\bar{Z}}(\mathfrak{U}_Z) = \mathbb{R}^{k \times (n-k)}$ is an open set. Finally, the map $\varphi_{\bar{Z}} \circ \varphi_{\bar{Z}}^{-1}$ from $\mathbb{R}^{(k-r) \times r}$ to $\mathbb{R}^{(k-r) \times r}$ is given by $\varphi_{\bar{Z}} \circ \varphi_{\bar{Z}}^{-1}(X) = \tilde{Z}_{\perp}^{+}(Z + Z_{\perp}X)G^{-1}$, with $G = \tilde{Z}^{+}(Z + Z_{\perp}X_Z)$, which is clearly an analytic map. $\square$

**Remark 1.** A possible choice for $Z_{\perp}$ satisfying $Z_{\perp}^T Z = 0$ is $Z_{\perp} = (id_k - ZZ^+)B_{\perp}$ where $B_{\perp} \in \mathcal{M}_{k-r}(\mathbb{R}^{k \times (k-r)})$ is such that its column space is a complement of the column space of $Z$. In practice, we can determine a set of $r$ linear independent rows of $Z$ (see, e.g., [25,26]), with indices $I$, and then choose $B_{\perp}$ such that $(B_{\perp})_{i,j} = \delta_{i,j}$ if $i \notin I$ and $0$ if $i \in I$, for $1 \le i \le k, 1 \le j \le k-r$. For a given $X \in \mathbb{R}^{(k-r) \times r}$, the computation of $Z_{\perp}X$ does not require $Z_{\perp}$ and has a complexity $O(r^2k)$.

## 2.2. Lie Group Structure of Neighbourhoods $\mathfrak{U}_Z$

Here we prove that each neighbourhood $\mathfrak{U}_Z$ of $G_r(\mathbb{R}^k)$ is a Lie group. For that, we first note that a neighbourhood $\mathfrak{U}_Z$ of $G_r(\mathbb{R}^k)$ can be identified with the set $S_Z$ through the application $\mathrm{col}_{k,r}: S_Z \to \mathfrak{U}_Z$. The next step is to identify $S_Z$ with a closed Lie subgroup of $\mathrm{GL}_k$, denoted by $G_Z$, with associated Lie algebra $g_Z$ isomorphic to $\mathbb{R}^{r \times (k-r)}$, and such that the exponential map $\exp: g_Z \to G_Z$ is a diffeomorphism. (We recall that the matrix exponential $\exp: \mathbb{R}^{k \times k} \to \mathrm{GL}_k$ is defined by $\exp(A) = \sum_{n=0}^{\infty} \frac{A^n}{n!}$.) To this end, for a given $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, we introduce the vector space

$$ g_Z := \{Z_{\perp}XZ^+ : X \in \mathbb{R}^{(k-r) \times r}\} \subset \mathbb{R}^{k \times k}. \qquad (5) $$

The following proposition proves that $g_Z$ is a commutative subalgebra of $\mathbb{R}^{k \times k}$.

**Proposition 2.** For all $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$,

$$ (Z_{\perp}XZ^{+})(Z_{\perp}\tilde{X}Z^{+}) = 0 $$

holds, and $g_Z$ is a commutative subalgebra of $\mathbb{R}^{k \times k}$. Moreover,

$$ \exp(Z_{\perp}XZ^{+}) = id_{k} + Z_{\perp}XZ^{+}, \qquad (6) $$

$$ \exp(Z_{\perp}XZ^{+})Z = Z + Z_{\perp}X, \qquad (7) $$

and

$$ \exp(Z_{\perp}XZ^{+})Z_{\perp} = Z_{\perp} \qquad (8) $$

hold for all $X \in \mathbb{R}^{(k-r) \times r}$.

**Proof.** Since $(Z_{\perp}XZ^{+})(Z_{\perp}\tilde{X}Z^{+}) = 0$ holds for all $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$, the vector space $g_Z$ is a closed subalgebra of the matrix unitary algebra $\mathbb{R}^{k \times k}$. As a consequence, $(Z_{\perp}XZ^{+})^p = 0$ holds for all $X \in \mathbb{R}^{(k-r) \times r}$ and all $p \ge 2$, which proves (6). We directly deduce (7) using $ZZ^{+} = id_r$ and (8) using $Z^{+}Z_{\perp} = 0$. $\square$

From Proposition 2 and the definition of $S_Z$, we obtain the following results.

**Corollary 2.** The affine cross section $S_Z$ satisfies

$$ S_Z = \{\exp(Z_{\perp}XZ^{+})Z : X \in \mathbb{R}^{(k-r) \times r}\}, \qquad (9) $$

and

$$ [\exp(Z_{\perp}XZ^{+})Z | Z_{\perp}] \in \mathrm{GL}_{k} \qquad (10) $$

for all $X \in \mathbb{R}^{(k-r) \times r}$, where the brackets $[\cdot|\cdot]$ are used for matrix concatenation.
**Proof.** From Proposition 2 and (4), we obtain (9) and we can write

$$[\exp(Z_{\perp}XZ^{+})Z|Z_{\perp}] = [\exp(Z_{\perp}XZ^{+})Z|\exp(Z_{\perp}XZ^{+})Z_{\perp}] = \exp(Z_{\perp}XZ^{+})[Z|Z_{\perp}].$$

Since $\exp(Z_{\perp}XZ^{+}), [Z|Z_{\perp}] \in GL_k$, (10) follows. □

Now we need to introduce the following definition and proposition (see ([27] p. 80)).

**Definition 1.** Let $(\mathbb{K}, +, \cdot)$ be a ring and let $(\mathbb{K}, +)$ be its additive group. A subset $\mathbb{I} \subset \mathbb{K}$ is called a two-sided ideal (or simply an ideal) of $\mathbb{K}$ if it is an additive subgroup of $\mathbb{K}$ such that $\mathbb{I} \cdot \mathbb{K} := \{r \cdot x : r \in \mathbb{I} \text{ and } x \in \mathbb{K}\} \subset \mathbb{I}$ and $\mathbb{K} \cdot \mathbb{I} := \{x \cdot r : r \in \mathbb{I} \text{ and } x \in \mathbb{K}\} \subset \mathbb{I}$.

**Proposition 3.** If $\mathfrak{g} \subset \mathfrak{h}$ is a two-sided ideal of the Lie algebra $\mathfrak{h}$ of a group $\mathcal{H}$, then the subgroup $\mathcal{G} \subset \mathcal{H}$ generated by $\exp(\mathfrak{g}) = \{\exp(G) : G \in \mathfrak{g}\}$ is normal and closed, with Lie algebra $\mathfrak{h}$.

From the above proposition, we deduce the following result.

**Lemma 2.** Let $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ and $Z_{\perp} \in \mathcal{M}_{k-r}(\mathbb{R}^{k \times (k-r)})$ be such that $Z^T Z_{\perp} = 0$. Then $\mathfrak{g}_Z \subset \mathbb{R}^{k \times k}$ is a two-sided ideal of the Lie algebra $\mathbb{R}^{k \times k}$ and hence

$$\mathcal{G}_Z := \{\exp(Z_{\perp}XZ^{+}) : X \in \mathbb{R}^{(k-r) \times r}\} \quad (11)$$

is a closed Lie group with Lie algebra $\mathfrak{g}_Z$. Furthermore, the map $\exp : \mathfrak{g}_Z \to \mathcal{G}_Z$ is bijective.

**Proof.** Consider $Z_{\perp}XZ^{+} \in \mathfrak{g}_Z$ and $A \in \mathbb{R}^{k \times k}$. Noting that $Z^{+}Z = \text{id}_r$ and $(Z_{\perp})^{+}Z_{\perp} = \text{id}_{k-r}$, we have that

$$(Z_{\perp}XZ^{+})A = Z_{\perp}(XZ^{+}AZ)Z^{+},$$

which proves that $\mathfrak{g}_Z \cdot \mathbb{R}^{k \times k} \subset \mathfrak{g}_Z$. Similarly, we have that

$$A(Z_{\perp}XZ^{+}) = Z_{\perp}((Z_{\perp})^{+}AZ_{\perp}X)Z^{+},$$

which proves that $\mathbb{R}^{k \times k} \cdot \mathfrak{g}_Z \subset \mathfrak{g}_Z$. This proves that $\mathfrak{g}_Z$ is a two-sided ideal. The map $\exp$ is clearly surjective. To prove that it is injective, we assume $\exp(Z_{\perp}XZ^{+}) = \exp(Z_{\perp}\tilde{X}Z^{+})$ for $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$. Then, from (6), we obtain $Z + Z_{\perp}X = Z + Z_{\perp}\tilde{X}$ and hence $\tilde{X} = \tilde{X}$, i.e., $Z_{\perp}XZ^{+} = Z_{\perp}\tilde{X}Z^{+}$ in $\mathfrak{g}_Z$. □

Finally, we can prove the following result.

**Theorem 2.** The set $S_Z$ together with the group operation $\times_Z$ defined by

$$\exp(Z_{\perp}XZ^{+})Z \times_Z \exp(Z_{\perp}\tilde{X}Z^{+})Z = \exp(Z_{\perp}(X + \tilde{X})Z^{+})Z \quad (12)$$

for $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$ is a Lie group.

**Proof.** To prove that it is a Lie group, we simply note that the multiplication and inversion maps

$$\mu : S_Z \times S_Z \to S_Z, (W, \tilde{W}) \mapsto \exp(Z_{\perp}(Z_{\perp}^{+}(W-Z) + Z_{\perp}^{+}(\tilde{W}-Z))Z^{+})Z$$

and

$$\delta : S_Z \to S_Z, W \mapsto \exp(-Z_{\perp}Z_{\perp}^{+}(W-Z)Z^{+})Z$$

are analytic. □

It follows that $\mathfrak{U}_Z$ can be identified with a Lie group through the map $\varphi_Z$.
**Proof.** From Proposition 2 and (4), we obtain (9) and we can write

$$[\exp(Z_{\perp}XZ^{+})Z|Z_{\perp}] = [\exp(Z_{\perp}XZ^{+})Z|\exp(Z_{\perp}XZ^{+})Z_{\perp}] = \exp(Z_{\perp}XZ^{+})[Z|Z_{\perp}].$$

Since $\exp(Z_{\perp}XZ^{+}), [Z|Z_{\perp}] \in GL_k$, (10) follows. □

Now we need to introduce the following definition and proposition (see ([27] p. 80)).

**Definition 1.** Let $(\mathbb{K}, +, \cdot)$ be a ring and let $(\mathbb{K}, +)$ be its additive group. A subset $\mathbb{I} \subset \mathbb{K}$ is called a two-sided ideal (or simply an ideal) of $\mathbb{K}$ if it is an additive subgroup of $\mathbb{K}$ such that $\mathbb{I} \cdot \mathbb{K} := \{r \cdot x : r \in \mathbb{I} \text{ and } x \in \mathbb{K}\} \subset \mathbb{I}$ and $\mathbb{K} \cdot \mathbb{I} := \{x \cdot r : r \in \mathbb{I} \text{ and } x \in \mathbb{K}\} \subset \mathbb{I}$.

**Proposition 3.** If $\mathfrak{g} \subset \mathfrak{h}$ is a two-sided ideal of the Lie algebra $\mathfrak{h}$ of a group $\mathcal{H}$, then the subgroup $\mathcal{G} \subset \mathcal{H}$ generated by $\exp(\mathfrak{g}) = \{\exp(G) : G \in \mathfrak{g}\}$ is normal and closed, with Lie algebra $\mathfrak{h}$.

From the above proposition, we deduce the following result.

**Lemma 2.** Let $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ and $Z_{\perp} \in \mathcal{M}_{k-r}(\mathbb{R}^{k \times (k-r)})$ be such that $Z^T Z_{\perp} = 0$. Then $\mathfrak{g}_Z \subset \mathbb{R}^{k \times k}$ is a two-sided ideal of the Lie algebra $\mathbb{R}^{k \times k}$ and hence

$$\mathcal{G}_Z := \{\exp(Z_{\perp}XZ^{+}) : X \in \mathbb{R}^{(k-r) \times r}\} \quad (11)$$

is a closed Lie group with Lie algebra $\mathfrak{g}_Z$. Furthermore, the map $\exp : \mathfrak{g}_Z \to \mathcal{G}_Z$ is bijective.

**Proof.** Consider $Z_{\perp}XZ^{+} \in \mathfrak{g}_Z$ and $A \in \mathbb{R}^{k \times k}$. Noting that $Z^{+}Z = \text{id}_r$ and $(Z_{\perp})^{+}Z_{\perp} = \text{id}_{k-r}$, we have that

$$(Z_{\perp}XZ^{+})A = Z_{\perp}(XZ^{+}AZ)Z^{+},$$

which proves that $\mathfrak{g}_Z \cdot \mathbb{R}^{k \times k} \subset \mathfrak{g}_Z$. Similarly, we have that

$$A(Z_{\perp}XZ^{+}) = Z_{\perp}((Z_{\perp})^{+}AZ_{\perp}X)Z^{+},$$

which proves that $\mathbb{R}^{k \times k} \cdot \mathfrak{g}_Z \subset \mathfrak{g}_Z$. This proves that $\mathfrak{g}_Z$ is a two-sided ideal. The map $\exp$ is clearly surjective. To prove that it is injective, we assume $\exp(Z_{\perp}XZ^{+}) = \exp(Z_{\perp}\tilde{X}Z^{+})$ for $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$. Then, from (6), we obtain $Z + Z_{\perp}X = Z + Z_{\perp}\tilde{X}$ and hence $\tilde{X} = \tilde{X}$, i.e., $Z_{\perp}XZ^{+} = Z_{\perp}\tilde{X}Z^{+}$ in $\mathfrak{g}_Z$. □

Finally, we can prove the following result.

**Theorem 2.** The set $S_Z$ together with the group operation $\times_Z$ defined by

$$\exp(Z_{\perp}XZ^{+})Z \times_Z \exp(Z_{\perp}\tilde{X}Z^{+})Z = \exp(Z_{\perp}(X + \tilde{X})Z^{+})Z \quad (12)$$

for $X, \tilde{X} \in \mathbb{R}^{(k-r) \times r}$ is a Lie group.

**Proof.** To prove that it is a Lie group, we simply note that the multiplication and inversion maps

$$\mu : S_Z \times S_Z \to S_Z, (W, \tilde{W}) \mapsto \exp(Z_{\perp}(Z_{\perp}^{+}(W-Z) + Z_{\perp}^{+}(\tilde{W}-Z))Z^{+})Z$$

and

$$\delta : S_Z \to S_Z, W \mapsto \exp(-Z_{\perp}Z_{\perp}^{+}(W-Z)Z^{+})Z$$

are analytic. □

It follows that $\mathfrak{U}_Z$ can be identified with a Lie group through the map $\varphi_Z$.
**Theorem 3.** Each neighbourhood $\mathfrak{U}_Z$ of $\mathfrak{G}_r(\mathbb{R}^k)$ together with the group operation $\circ_Z$ defined by

$$ \mathcal{V} \circ_Z \mathcal{V}' = \varphi_Z^{-1}(\varphi_Z(\mathcal{V}) + \varphi_Z(\mathcal{V}')) $$

for $\mathcal{V}, \mathcal{V}' \in \mathfrak{U}_Z$, is a Lie group, and the map $\gamma_Z : \mathfrak{U}_Z \to \mathcal{G}_Z$ given by

$$ \gamma_Z(\mathcal{U}) = \exp(Z_{\perp} \varphi_Z(\mathcal{U}) Z^+) $$

is a Lie group isomorphism.

**3. The Non-Compact Stiefel Principal Bundle $M_r(\mathbb{R}^{k \times r})$**

In this section, we give a new geometric description of the set $M_r(\mathbb{R}^{k \times r})$ of matrices with full rank $r < k$, which is based on the geometric description of the Grassmann manifold given in Section 2.

**3.1. Principal Bundle Structure of $M_r(\mathbb{R}^{k \times r})$**

For $Z \in M_r(\mathbb{R}^{k \times r})$, we define a neighbourhood of Z as

$$ \mathcal{V}_Z := \{ W \in M_r(\mathbb{R}^{k \times r}) : \det(Z^T W) \neq 0 \} \supset \mathcal{S}_Z. \quad (13) $$

From Proposition 1, we know that for a given matrix $W \in \mathcal{V}_Z$, there exists a unique pair of matrices $(X, G) \in \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ such that $W = (Z + Z_{\perp}X)G$. Therefore,

$$ \mathcal{V}_Z = \{(Z + Z_{\perp}X)G : X \in \mathbb{R}^{(k-r) \times r}, G \in \text{GL}_r\}. $$

It allows us to introduce a parametrisation $\xi_Z^{-1}$ (see Figure 1) defined through the bijection

$$ \xi_Z : \mathcal{V}_Z \to \mathbb{R}^{(k-r) \times r} \times \text{GL}_r, \quad (14) $$

such that

$$ \xi_Z^{-1}(X, G) = (Z + Z_{\perp}X)G $$

for $(X, G) \in \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ and

$$ \xi_Z(W) = (Z_{\perp}^+ W (Z^+ W)^{-1}, Z^+ W) $$

for $W \in \mathcal{V}_Z$. In particular,

$$ \xi_Z^{-1}(0, \text{id}_r) = Z. $$

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Illustration of the chart $\xi_Z$ which associates with $W = (Z + Z_{\perp}X)G \in \mathcal{V}_Z \subset M_r(\mathbb{R}^{k \times r})$, the parameters $(X, G)$ in $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$.</figcaption></figure>

**Theorem 4.** The collection $\mathcal{B}_{k,r} := \{(\mathcal{V}_Z, \xi_Z) : Z \in M_r(\mathbb{R}^{k \times r})\}$ is an analytic atlas for $M_r(\mathbb{R}^{k \times r})$, and hence $(M_r(\mathbb{R}^{k \times r}), \mathcal{B}_{k,r})$ is an analytic kr-dimensional manifold modelled on $\mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r}$.
**Theorem 3.** Each neighbourhood $\mathfrak{U}_Z$ of $\mathfrak{G}_r(\mathbb{R}^k)$ together with the group operation $\circ_Z$ defined by

$$ \mathcal{V} \circ_Z \mathcal{V}' = \varphi_Z^{-1}(\varphi_Z(\mathcal{V}) + \varphi_Z(\mathcal{V}')) $$

for $\mathcal{V}, \mathcal{V}' \in \mathfrak{U}_Z$, is a Lie group, and the map $\gamma_Z : \mathfrak{U}_Z \to \mathcal{G}_Z$ given by

$$ \gamma_Z(\mathcal{U}) = \exp(Z_{\perp} \varphi_Z(\mathcal{U}) Z^{+}) $$

is a Lie group isomorphism.

### 3. The Non-Compact Stiefel Principal Bundle $\mathcal{M}_r(\mathbb{R}^{k \times r})$

In this section, we give a new geometric description of the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ of matrices with full rank $r < k$, which is based on the geometric description of the Grassmann manifold given in Section 2.

#### 3.1. Principal Bundle Structure of $\mathcal{M}_r(\mathbb{R}^{k \times r})$

For $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, we define a neighbourhood of Z as

$$ \mathcal{V}_Z := \{ W \in \mathcal{M}_r(\mathbb{R}^{k \times r}) : \det(Z^T W) \neq 0 \} \supset \mathcal{S}_Z. \qquad (13) $$

From Proposition 1, we know that for a given matrix $W \in \mathcal{V}_Z$, there exists a unique pair of matrices $(X, G) \in \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ such that $W = (Z + Z_{\perp}X)G$. Therefore,

$$ \mathcal{V}_Z = \{(Z + Z_{\perp}X)G : X \in \mathbb{R}^{(k-r) \times r}, G \in \text{GL}_r\}. $$

It allows us to introduce a parametrisation $\xi_Z^{-1}$ (see Figure 1) defined through the bijection

$$ \xi_Z : \mathcal{V}_Z \to \mathbb{R}^{(k-r) \times r} \times \text{GL}_r, \qquad (14) $$

such that

$$ \xi_Z^{-1}(X, G) = (Z + Z_{\perp}X)G $$

for $(X, G) \in \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ and

$$ \xi_Z(W) = (Z_{\perp}^+ W (Z^+ W)^{-1}, Z^+ W) $$

for $W \in \mathcal{V}_Z$. In particular,

$$ \xi_Z^{-1}(0, \text{id}_r) = Z. $$

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Illustration of the chart $\xi_Z$ which associates with $W = (Z + Z_{\perp}X)G \in \mathcal{V}_Z \subset \mathcal{M}_r(\mathbb{R}^{k \times r})$, the parameters $(X, G)$ in $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$.</figcaption></figure>

**Theorem 4.** The collection $\mathcal{B}_{k,r} := \{(\mathcal{V}_Z, \xi_Z) : Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})\}$ is an analytic atlas for $\mathcal{M}_r(\mathbb{R}^{k \times r})$, and hence $(\mathcal{M}_r(\mathbb{R}^{k \times r}), \mathcal{B}_{k,r})$ is an analytic kr-dimensional manifold modelled on $\mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r}$.
**Proof.** {$V_Z$}$_{Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})}$ is clearly a covering of $\mathcal{M}_r(\mathbb{R}^{k \times r})$. Moreover, since $\xi_Z$ is bijective from $V_Z$ to $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ we claim that if $V_Z \cap V_{\bar{Z}} \neq \emptyset$ for $Z, \bar{Z} \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, then the following statements hold:

(i) $\xi_Z(V_Z \cap V_{\bar{Z}})$ and $\xi_{\bar{Z}}(V_Z \cap V_{\bar{Z}})$ are open sets in $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ and

(ii) the map $\xi_{\bar{Z}} \circ \xi_Z^{-1}$ is analytic from $\xi_Z(V_Z \cap V_{\bar{Z}}) \subset \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ to $\xi_{\bar{Z}}(V_Z \cap V_{\bar{Z}}) \subset \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$.

In this proof, we equip $\mathbb{R}^{k \times r}$ with the topology $\tau_{\mathbb{R}^{k \times r}}$ induced by matrix norms. For any $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, $V_Z = \{W \in \mathbb{R}^{k \times r} : \det(Z^T W) \neq 0\}$ is the inverse image of the open set $\mathbb{R} \setminus \{0\}$ by the continuous map $W \mapsto \det(Z^T W)$ from $\mathbb{R}^{k \times r}$ to $\mathbb{R}$, and therefore, $V_Z$ is an open set of $\mathbb{R}^{k \times r}$. Since $V_Z$ and $V_{\bar{Z}}$ are open sets in $\mathbb{R}^{k \times r}$, $V_Z \cap V_{\bar{Z}}$ is also an open set in $\mathbb{R}^{k \times r}$ and since $\xi_Z^{-1}$ is a continuous map from $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ to $\mathbb{R}^{k \times r}$, the set $\xi_Z(V_Z \cap V_{\bar{Z}})$, as the inverse image of an open set by a continuous map, is an open set in $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$. Similarly, $\xi_{\bar{Z}}(V_Z \cap V_{\bar{Z}})$ is an open set. Now let $(X, G) \in \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ such that $\xi_Z^{-1}(X, G) \in V_Z \cap V_{\bar{Z}}$. From the expressions of $\xi_Z^{-1}$ and $\xi_{\bar{Z}}$, the map $\xi_Z \circ \xi_Z^{-1}$ is defined by

$$ \xi_{\bar{Z}} \circ \xi_Z^{-1}(X, G) = (\tilde{Z}_{\perp}^{+} \xi_Z^{-1}(X, G)(\tilde{Z}^{+} \xi_Z^{-1}(X, G))^{-1}, \tilde{Z}^{+} \xi_Z^{-1}(X, G)), $$

with $\xi_Z^{-1}(X, G) = (Z + Z_{\perp} X)G$, which is clearly an analytic map. $\square$

Before stating the next result, we recall the definition of a morphism between manifolds and of a fibre bundle. We introduce notions of $C^p$ maps and $C^p$ manifolds, with $p \in \mathbb{N} \cup \{\infty\}$ or $p = \omega$. In the latter case, $C \to$ means analytic.

**Definition 2.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be two $C^p$ manifolds. Let $F : \mathbb{M} \to \mathbb{N}$ be a map. We say that $F$ is a $C^p$ morphism between $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ if given $m \in \mathbb{M}$, there exists a chart $(U, \varphi) \in \mathcal{A}$ such that $m \in U$ and a chart $(W, \psi) \in \mathcal{B}$ such that $F(m) \in W$ where $F(U) \subset W$, and the map

$$ \psi \circ F \circ \varphi^{-1} : \varphi(U) \to \psi(W) $$

is a map of class $C^p$. If it is a $C^p$ diffeomorphism, then we say that $F$ is a $C^p$ diffeomorphism between manifolds. We say that $\psi \circ F \circ \varphi^{-1}$ is a representation of $F$ using a system of local coordinates given by the charts $(U, \varphi)$ and $(W, \psi)$.

**Definition 3.** Let $\mathbb{B}$ be a $C^p$ manifold with atlas $\mathcal{A} = \{(U_b, \varphi_b) : b \in \mathbb{B}\}$, and let $\mathbb{F}$ be a manifold. A $C^p$ fibre bundle $\mathbb{E}$ with base $\mathbb{B}$ and typical fibre $\mathbb{F}$ is a $C^p$ manifold which is locally a product manifold; that is, there exists a surjective morphism $\pi : \mathbb{E} \to \mathbb{B}$ such that for each $b \in \mathbb{B}$ there is a $C^p$ diffeomorphism between manifolds

$$ \chi_b : \pi^{-1}(U_b) \to U_b \times \mathbb{F}, $$

such that $p_b \circ \chi_b = \pi$ where $p_b : U_b \times \mathbb{F} \to U_b$ is the projection. For each $b \in \mathbb{B}$, $\pi^{-1}(b) = E_b$ is called the fibre over $b$. The $C^p$ diffeomorphisms $\chi_b$ are called fibre bundle charts. If $p = 0$, $\mathbb{E}$, $\mathbb{B}$ and $\mathbb{F}$ are only required to be topological spaces and $\{U_b : b \in \mathbb{B}\}$ an open covering of $\mathbb{B}$. In the case where $\mathbb{F}$ is a Lie group, we say that $\mathbb{E}$ is a $C^p$ principal bundle, and if $\mathbb{F}$ is a vector space, we say that it is a $C^p$ vector bundle.

**Theorem 5.** The set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ is an analytic principal bundle with typical fibre $\text{GL}_r$ and base $\text{G}_r(\mathbb{R}^k)$, with a surjective morphism between $\mathcal{M}_r(\mathbb{R}^{k \times r})$ and $\text{G}_r(\mathbb{R}^k)$ given by the map $\text{col}_{k,r}$.

**Proof.** To show that it is an analytic principal bundle, we first observe that

$$ \text{col}_{k,r} : (\mathcal{M}_r(\mathbb{R}^{k \times r}), \mathcal{B}_{k,r}) \to (\mathbb{G}_r(\mathbb{R}^k), \mathcal{A}_{k,r}) $$

is a surjective morphism. Indeed, let $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ and $(V_Z, \xi_Z) \in \mathcal{B}_{k,r}$ and $(\mathfrak{U}_Z, \varphi_Z) \in \mathcal{A}_{k,r}$. Noting that $\text{col}_{k,r}(\mathbb{Y}G) = \text{col}_{k,r}(\mathbb{Y})$ for all $\mathbb{Y} \in S_Z$, we obtain that $\text{col}_{k,r}(V_Z) = \mathfrak{U}_Z$.
**Proof.** {$V_Z$}$_{Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})}$ is clearly a covering of $\mathcal{M}_r(\mathbb{R}^{k \times r})$. Moreover, since $\xi_Z$ is bijective from $V_Z$ to $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ we claim that if $V_Z \cap V_{\bar{Z}} \neq \emptyset$ for $Z, \bar{Z} \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, then the following statements hold:

(i) $\xi_Z(V_Z \cap V_{\bar{Z}})$ and $\xi_{\bar{Z}}(V_Z \cap V_{\bar{Z}})$ are open sets in $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ and

(ii) the map $\xi_{\bar{Z}} \circ \xi_Z^{-1}$ is analytic from $\xi_Z(V_Z \cap V_{\bar{Z}}) \subset \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ to $\xi_{\bar{Z}}(V_Z \cap V_{\bar{Z}}) \subset \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$.

In this proof, we equip $\mathbb{R}^{k \times r}$ with the topology $\tau_{\mathbb{R}^{k \times r}}$ induced by matrix norms. For any $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$, $V_Z = \{W \in \mathbb{R}^{k \times r} : \det(Z^T W) \neq 0\}$ is the inverse image of the open set $\mathbb{R} \setminus \{0\}$ by the continuous map $W \mapsto \det(Z^T W)$ from $\mathbb{R}^{k \times r}$ to $\mathbb{R}$, and therefore, $V_Z$ is an open set of $\mathbb{R}^{k \times r}$. Since $V_Z$ and $V_{\bar{Z}}$ are open sets in $\mathbb{R}^{k \times r}$, $V_Z \cap V_{\bar{Z}}$ is also an open set in $\mathbb{R}^{k \times r}$ and since $\xi_Z^{-1}$ is a continuous map from $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ to $\mathbb{R}^{k \times r}$, the set $\xi_Z(V_Z \cap V_{\bar{Z}})$, as the inverse image of an open set by a continuous map, is an open set in $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$. Similarly, $\xi_{\bar{Z}}(V_Z \cap V_{\bar{Z}})$ is an open set. Now let $(X, G) \in \mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ such that $\xi_Z^{-1}(X, G) \in V_Z \cap V_{\bar{Z}}$. From the expressions of $\xi_Z^{-1}$ and $\xi_{\bar{Z}}$, the map $\xi_Z \circ \xi_Z^{-1}$ is defined by

$$ \xi_{\bar{Z}} \circ \xi_Z^{-1}(X, G) = (\tilde{Z}_{\perp}^{+} \xi_Z^{-1}(X, G)(\tilde{Z}^{+} \xi_Z^{-1}(X, G))^{-1}, \tilde{Z}^{+} \xi_Z^{-1}(X, G)), $$

with $\xi_Z^{-1}(X, G) = (Z + Z_{\perp} X)G$, which is clearly an analytic map. $\square$

Before stating the next result, we recall the definition of a morphism between manifolds and of a fibre bundle. We introduce notions of $C^p$ maps and $C^p$ manifolds, with $p \in \mathbb{N} \cup \{\infty\}$ or $p = \omega$. In the latter case, $C \to$ means analytic.

**Definition 2.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be two $C^p$ manifolds. Let $F : \mathbb{M} \to \mathbb{N}$ be a map. We say that $F$ is a $C^p$ morphism between $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ if given $m \in \mathbb{M}$, there exists a chart $(U, \varphi) \in \mathcal{A}$ such that $m \in U$ and a chart $(W, \psi) \in \mathcal{B}$ such that $F(m) \in W$ where $F(U) \subset W$, and the map

$$ \psi \circ F \circ \varphi^{-1} : \varphi(U) \to \psi(W) $$

is a map of class $C^p$. If it is a $C^p$ diffeomorphism, then we say that $F$ is a $C^p$ diffeomorphism between manifolds. We say that $\psi \circ F \circ \varphi^{-1}$ is a representation of $F$ using a system of local coordinates given by the charts $(U, \varphi)$ and $(W, \psi)$.

**Definition 3.** Let $\mathbb{B}$ be a $C^p$ manifold with atlas $\mathcal{A} = \{(U_b, \varphi_b) : b \in \mathbb{B}\}$, and let $\mathbb{F}$ be a manifold. A $C^p$ fibre bundle $\mathbb{E}$ with base $\mathbb{B}$ and typical fibre $\mathbb{F}$ is a $C^p$ manifold which is locally a product manifold; that is, there exists a surjective morphism $\pi : \mathbb{E} \to \mathbb{B}$ such that for each $b \in \mathbb{B}$ there is a $C^p$ diffeomorphism between manifolds

$$ \chi_b : \pi^{-1}(U_b) \to U_b \times \mathbb{F}, $$

such that $p_b \circ \chi_b = \pi$ where $p_b : U_b \times \mathbb{F} \to U_b$ is the projection. For each $b \in \mathbb{B}$, $\pi^{-1}(b) = E_b$ is called the fibre over $b$. The $C^p$ diffeomorphisms $\chi_b$ are called fibre bundle charts. If $p = 0$, $E, \mathbb{B}$ and $\mathbb{F}$ are only required to be topological spaces and $\{U_b : b \in \mathbb{B}\}$ an open covering of $\mathbb{B}$. In the case where $\mathbb{F}$ is a Lie group, we say that $E$ is a $C^p$ principal bundle, and if $\mathbb{F}$ is a vector space, we say that it is a $C^p$ vector bundle.

**Theorem 5.** The set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ is an analytic principal bundle with typical fibre $\text{GL}_r$ and base $\text{G}_r(\mathbb{R}^k)$, with a surjective morphism between $\mathcal{M}_r(\mathbb{R}^{k \times r})$ and $\text{G}_r(\mathbb{R}^k)$ given by the map $\text{col}_{k,r}$.

**Proof.** To show that it is an analytic principal bundle, we first observe that

$$ \text{col}_{k,r} : (\mathcal{M}_r(\mathbb{R}^{k \times r}), \mathcal{B}_{k,r}) \to (\mathbb{G}_r(\mathbb{R}^k), \mathcal{A}_{k,r}) $$

is a surjective morphism. Indeed, let $Z \in \mathcal{M}_r(\mathbb{R}^{k \times r})$ and $(V_Z, \xi_Z) \in \mathcal{B}_{k,r}$ and $(\mathfrak{U}_Z, \varphi_Z) \in \mathcal{A}_{k,r}$. Noting that $\text{col}_{k,r}(YG) = \text{col}_{k,r}(Y)$ for all $Y \in S_Z$, we obtain that $\text{col}_{k,r}(V_Z) = \mathfrak{U}_Z$.
Moreover, a representation of $\text{col}_{k,r}$ by using a system of local coordinates given by the charts is

$$ (\varphi_Z \circ \text{col}_{k,r} \circ \xi_Z^{-1})(X, G) = X, $$

which is clearly an analytic map from $\mathbb{R}^{(k-r) \times r} \times \text{GL}_r$ to $\mathbb{R}^{(k-r) \times r}$ such that $\text{col}_{k,r}^{-1}(\mathfrak{U}_Z) = \mathcal{V}_Z$. Now, a representation of the morphism

$$ \chi_Z : (\mathcal{V}_Z, \{(\mathcal{V}_Z, \xi_Z)\}) \to (\mathfrak{U}_Z, \{(\mathfrak{U}_Z, \varphi_Z)\}) \times (\text{GL}_r, \{(\text{GL}_r, \text{id}_{\mathbb{R}^{k \times r}})\}), \quad W \mapsto (\text{col}_{k,r}(W), G) $$

using the system of local coordinates given by the charts is

$$ ((\varphi_Z \times \text{id}_{\mathbb{R}^{k \times r}}) \circ \chi_Z \circ \xi_Z^{-1}) : \mathbb{R}^{(k-r) \times r} \times \text{GL}_r \to \mathbb{R}^{(k-r) \times r} \times \text{GL}_r, $$

defined by

$$ ((\varphi_Z \times \text{id}_{\mathbb{R}^{k \times r}}) \circ \chi_Z \circ \xi_Z^{-1})(X, G) = (X, G), $$

which is clearly an analytic diffeomorphism. To conclude, consider the projection

$$ p_Z : \mathfrak{U}_Z \times \text{GL}_r \to \mathfrak{U}_Z, \quad (\mathfrak{U}, G) \mapsto \mathfrak{U}, $$

and observe that $(p_Z \circ \chi_Z)(W) = \text{col}_{k,r}(W)$ holds for all $W \in \mathcal{V}_Z$. $\square$

### 3.2. $\mathcal{M}_r(\mathbb{R}^{k \times r})$ as a Submanifold and Its Tangent Space

Here, we prove that the non-compact Stiefel manifold $\mathcal{M}_r(\mathbb{R}^{k \times r})$ equipped with the topology given by the atlas $\mathcal{B}_{k,r}$ is an embedded submanifold in $\mathbb{R}^{k \times r}$. For that, we have to prove that the standard inclusion map

$$ i : (\mathcal{M}_r(\mathbb{R}^{k \times r}), \mathcal{B}_{k,r}) \to (\mathbb{R}^{k \times r}, \{(\mathbb{R}^{k \times r}, \text{id}_{\mathbb{R}^{k \times r}})\}) $$

as a morphism is an embedding. To see this, we need to recall some definitions and results.

**Definition 4.** Let $F : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a morphism between $\mathcal{C}^p$ manifolds and let $m \in \mathbb{M}$. We say that $F$ is an immersion at $m$ if there exists an open neighbourhood $U_m$ of $m$ in $\mathbb{M}$ such that the restriction of $F$ to $U_m$ induces an isomorphism from $U_m$ onto a submanifold of $\mathbb{N}$. We say that $F$ is an immersion if it is an immersion at each point of $\mathbb{M}$.

The next step is to recall the definition of the differential as a morphism which gives a linear map between the tangent spaces of the manifolds (in local coordinates) involved with the morphism. Let us recall that for any $m \in \mathbb{M}$, we denote by $T_m\mathbb{M}$ the tangent space of $\mathbb{M}$ at $m$ (in local coordinates).

**Definition 5.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be two $\mathcal{C}^p$ manifolds. Let $F : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a morphism of class $\mathcal{C}^p$; i.e., for any $m \in \mathbb{M}$,

$$ \psi \circ F \circ \varphi^{-1} : \varphi(U) \to \psi(W) $$

is a map of class $\mathcal{C}^p$, where $(U, \varphi) \in \mathcal{A}$ is a chart in $\mathbb{M}$ containing $m$ and $(W, \psi) \in \mathcal{B}$ is a chart in $\mathbb{N}$ containing $F(m)$. Then we define

$$ T_m F : T_m(\mathbb{M}) \to T_{F(m)}(\mathbb{N}), \quad v \mapsto D(\psi \circ F \circ \varphi^{-1})(\varphi(m))[v]. $$

For finite dimensional manifolds we have the following criterion for immersions (see Theorem 3.5.7 in [28]).

**Proposition 4.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be $\mathcal{C}^p$ manifolds. Let

$$ F : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B}) $$
Moreover, a representation of $\mathrm{col}_{k,r}$ by using a system of local coordinates given by the charts is

$$ (\varphi_Z \circ \mathrm{col}_{k,r} \circ \xi_Z^{-1})(X, G) = X, $$

which is clearly an analytic map from $\mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r$ to $\mathbb{R}^{(k-r) \times r}$ such that $\mathrm{col}_{k,r}^{-1}(\mathfrak{U}_Z) = \mathcal{V}_Z$. Now, a representation of the morphism

$$ \chi_Z : (\mathcal{V}_Z, \{(\mathcal{V}_Z, \xi_Z)\}) \to (\mathfrak{U}_Z, \{(\mathfrak{U}_Z, \varphi_Z)\}) \times (\mathrm{GL}_r, \{(\mathrm{GL}_r, \mathrm{id}_{\mathbb{R}^{k \times r}})\}), \quad W \mapsto (\mathrm{col}_{k,r}(W), G) $$

using the system of local coordinates given by the charts is

$$ ((\varphi_Z \times \mathrm{id}_{\mathbb{R}^{k \times r}}) \circ \chi_Z \circ \xi_Z^{-1}) : \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r \to \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r, $$

defined by

$$ ((\varphi_Z \times \mathrm{id}_{\mathbb{R}^{k \times r}}) \circ \chi_Z \circ \xi_Z^{-1})(X, G) = (X, G), $$

which is clearly an analytic diffeomorphism. To conclude, consider the projection

$$ p_Z : \mathfrak{U}_Z \times \mathrm{GL}_r \to \mathfrak{U}_Z, \quad (\mathfrak{U}, G) \mapsto \mathfrak{U}, $$

and observe that $(p_Z \circ \chi_Z)(W) = \mathrm{col}_{k,r}(W)$ holds for all $W \in \mathcal{V}_Z$. $\square$

### 3.2. $\mathcal{M}_r(\mathbb{R}^{k \times r})$ as a Submanifold and Its Tangent Space

Here, we prove that the non-compact Stiefel manifold $\mathcal{M}_r(\mathbb{R}^{k \times r})$ equipped with the topology given by the atlas $\mathcal{B}_{k,r}$ is an embedded submanifold in $\mathbb{R}^{k \times r}$. For that, we have to prove that the standard inclusion map

$$ i : (\mathcal{M}_r(\mathbb{R}^{k \times r}), \mathcal{B}_{k,r}) \to (\mathbb{R}^{k \times r}, \{(\mathbb{R}^{k \times r}, \mathrm{id}_{\mathbb{R}^{k \times r}})\}) $$

as a morphism is an embedding. To see this, we need to recall some definitions and results.

**Definition 4.** Let $F : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a morphism between $\mathcal{C}^p$ manifolds and let $m \in \mathbb{M}$. We say that $F$ is an immersion at $m$ if there exists an open neighbourhood $U_m$ of $m$ in $\mathbb{M}$ such that the restriction of $F$ to $U_m$ induces an isomorphism from $U_m$ onto a submanifold of $\mathbb{N}$. We say that $F$ is an immersion if it is an immersion at each point of $\mathbb{M}$.

The next step is to recall the definition of the differential as a morphism which gives a linear map between the tangent spaces of the manifolds (in local coordinates) involved with the morphism. Let us recall that for any $m \in \mathbb{M}$, we denote by $\mathbb{T}_m \mathbb{M}$ the tangent space of $\mathbb{M}$ at $m$ (in local coordinates).

**Definition 5.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be two $\mathcal{C}^p$ manifolds. Let $F : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a morphism of class $\mathcal{C}^p$; i.e., for any $m \in \mathbb{M}$,

$$ \psi \circ F \circ \varphi^{-1} : \varphi(U) \to \psi(W) $$

is a map of class $\mathcal{C}^p$, where $(U, \varphi) \in \mathcal{A}$ is a chart in $\mathbb{M}$ containing $m$ and $(W, \psi) \in \mathcal{B}$ is a chart in $\mathbb{N}$ containing $F(m)$. Then we define

$$ \mathbb{T}_m F : \mathbb{T}_m(\mathbb{M}) \to \mathbb{T}_{F(m)}(\mathbb{N}), \quad v \mapsto D(\psi \circ F \circ \varphi^{-1})(\varphi(m))[v]. $$

For finite dimensional manifolds we have the following criterion for immersions (see Theorem 3.5.7 in [28]).

**Proposition 4.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be $\mathcal{C}^p$ manifolds. Let

$$ F : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B}) $$
be a $C^p$ morphism and $m \in \mathbb{M}$. Then $F$ is an immersion at $m$ if and only if $T_m F$ is injective.

A concept related to an immersion between manifolds is given in the following definition.

**Definition 6.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be $C^p$ manifolds and let $f : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a $C^p$ morphism. If $f$ is an injective immersion, then $f(\mathbb{M})$ is called an immersed submanifold of $\mathbb{N}$.

Finally, we give the definition of embedding.

**Definition 7.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be $C^p$ manifolds and let $f : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a $C^p$ morphism. If $f$ is an injective immersion, and $f : (\mathbb{M}, \tau_{\mathcal{A}}) \to (f(\mathbb{M}), \tau_{\mathcal{B}}|_{f(\mathbb{M})})$ is a topological homeomorphism, then we say that $f$ is an embedding and $f(\mathbb{M})$ is called an embedded submanifold of $\mathbb{N}$.

We first note that the representation of the inclusion map $i$ using the system of local coordinates given by the charts $(\mathcal{V}_Z, \xi_Z) \in \mathcal{B}_{k,r}$ in $M_r(\mathbb{R}^{k \times r})$ and $(\mathbb{R}^{k \times r}, \mathrm{id}_{\mathbb{R}^{k \times r}})$ in $\mathbb{R}^{k \times r}$ is

$$(\mathrm{id}_{\mathbb{R}^{k \times r}} \circ i \circ \xi_Z^{-1}) = (i \circ \xi_Z^{-1}) : \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r \to \mathbb{R}^{k \times r}, \quad (X, G) \mapsto (Z + Z_{\perp} X)G.$$

Then the tangent map $T_{Zi}$ at $Z = \xi_Z^{-1}(0, id_r)$, defined by $T_{Zi} = D(i \circ \xi_Z^{-1})(0, id_r)$, is

$$T_{Zi} : \mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{k \times r}, \quad (\dot{X}, \dot{G}) \mapsto Z_{\perp} \dot{X} + Z \dot{G}.$$

**Proposition 5.** The tangent map $T_{Zi} : \mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{k \times r}$ at $Z \in M_r(\mathbb{R}^{k \times r})$ is a linear isomorphism, with inverse $(T_{Zi})^{-1}$ given by

$$(T_{Zi})^{-1}(\dot{Z}) = (Z_{\perp}^{+} \dot{Z}, Z^{+} \dot{Z}),$$

for $\dot{Z} \in \mathbb{R}^{k \times r}$. Furthermore, the standard inclusion map $i$ is an embedding from $M_r(\mathbb{R}^{k \times r})$ to $\mathbb{R}^{k \times r}$.

**Proof.** Let us assume that $T_{Zi}(\dot{X}, \dot{G}) = Z_{\perp} \dot{X} + Z \dot{G} = 0$. Multiplying this equality by $Z^+$ and $Z_{\perp}^+$ on the left, we obtain $\dot{G} = 0$ and $\dot{X} = 0$, respectively, which implies that $T_{Zi}$ is injective. To prove that it is also surjective, we consider a matrix $\dot{Z} \in \mathbb{R}^{k \times r}$ and observe that $\dot{X} = Z_{\perp}^{+} \dot{Z} \in \mathbb{R}^{(k-r) \times r}$ and $\dot{G} = Z^{+} \dot{Z} \in \mathbb{R}^{r \times r}$ is such that $T_{Zi}(\dot{X}, \dot{G}) = \dot{Z}$. Since $T_{Zi}$ is injective, the inclusion map $i$ is an immersion.

To prove that it is an embedding, we equip $M_r(\mathbb{R}^{k \times r})$ with the topology $\tau_{\mathcal{B}_{k,r}}$ given by the atlas and we equip $\mathbb{R}^{k \times r}$ with the topology $\tau_{\mathbb{R}^{k \times r}}$ induced by matrix norms. We need to check that

$$i : (M_r(\mathbb{R}^{k \times r}), \tau_{\mathcal{B}_{k,r}}) \to (M_r(\mathbb{R}^{k \times r}), \tau_{\mathbb{R}^{k \times r}|_{M_r(\mathbb{R}^{k \times r})})$$

is a topological homeomorphism. Since the topology in $(M_r(\mathbb{R}^{k \times r}), \tau_{\mathcal{B}_{k,r}})$ has the property that each local chart $\xi_Z$ is indeed a homeomorphism from $\mathcal{V}_Z$ in $M_r(\mathbb{R}^{k \times r})$ to $\xi_Z(\mathcal{V}_Z) = \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r$ (see Section 1.1), we only need to show that the bijection $(i \circ \xi_Z^{-1}) : \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r \to \mathcal{V}_Z \subset \mathbb{R}^{k \times r}$ given by

$$(i \circ \xi_Z^{-1})(X, G) = (Z + Z_{\perp} X)G$$

is a topological homeomorphism for all $Z \in M_r(\mathbb{R}^{k \times r})$. Observe that $D(i \circ \xi_Z^{-1})(X, G) \in \mathcal{L}(\mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r}, \mathbb{R}^{k \times r})$ is given by

$$D(i \circ \xi_Z^{-1})(X, G)[(\dot{X}, \dot{G})] = Z_{\perp} \dot{X} G + (Z + Z_{\perp} X) \dot{G}.$$

Assume that $Z_{\perp} \dot{X} G + (Z + Z_{\perp} X) \dot{G} = 0$. Multiplying this equality by $Z^+$ on the left we obtain $\dot{G} = 0$, and hence $Z_{\perp} \dot{X} G = 0$. Multiplying by $Z_{\perp}^+$ on the left, we obtain
be a $C^p$ morphism and $m \in \mathbb{M}$. Then $F$ is an immersion at $m$ if and only if $T_m F$ is injective.

A concept related to an immersion between manifolds is given in the following definition.

**Definition 6.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be $C^p$ manifolds and let $f : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a $C^p$ morphism. If $f$ is an injective immersion, then $f(\mathbb{M})$ is called an immersed submanifold of $\mathbb{N}$.

Finally, we give the definition of embedding.

**Definition 7.** Let $(\mathbb{M}, \mathcal{A})$ and $(\mathbb{N}, \mathcal{B})$ be $C^p$ manifolds and let $f : (\mathbb{M}, \mathcal{A}) \to (\mathbb{N}, \mathcal{B})$ be a $C^p$ morphism. If $f$ is an injective immersion, and $f : (\mathbb{M}, \tau_{\mathcal{A}}) \to (f(\mathbb{M}), \tau_{\mathcal{B}}|_{f(\mathbb{M})})$ is a topological homeomorphism, then we say that $f$ is an embedding and $f(\mathbb{M})$ is called an embedded submanifold of $\mathbb{N}$.

We first note that the representation of the inclusion map $i$ using the system of local coordinates given by the charts $(\mathcal{V}_Z, \xi_Z) \in \mathcal{B}_{k,r}$ in $M_r(\mathbb{R}^{k \times r})$ and $(\mathbb{R}^{k \times r}, \mathrm{id}_{\mathbb{R}^{k \times r}})$ in $\mathbb{R}^{k \times r}$ is

$$(\mathrm{id}_{\mathbb{R}^{k \times r}} \circ i \circ \xi_Z^{-1}) = (i \circ \xi_Z^{-1}) : \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r \to \mathbb{R}^{k \times r}, \quad (X, G) \mapsto (Z + Z_{\perp} X)G.$$

Then the tangent map $T_{Zi}$ at $Z = \xi_Z^{-1}(0, \mathrm{id}_r)$, defined by $T_{Zi} = D(i \circ \xi_Z^{-1})(0, \mathrm{id}_r)$, is

$$T_{Zi} : \mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{k \times r}, \quad (\dot{X}, \dot{G}) \mapsto Z_{\perp} \dot{X} + Z \dot{G}.$$

**Proposition 5.** The tangent map $T_{Zi} : \mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{k \times r}$ at $Z \in M_r(\mathbb{R}^{k \times r})$ is a linear isomorphism, with inverse $(T_{Zi})^{-1}$ given by

$$(T_{Zi})^{-1}(\dot{Z}) = (Z_{\perp}^{+} \dot{Z}, Z^{+} \dot{Z}),$$

for $\dot{Z} \in \mathbb{R}^{k \times r}$. Furthermore, the standard inclusion map $i$ is an embedding from $M_r(\mathbb{R}^{k \times r})$ to $\mathbb{R}^{k \times r}$.

**Proof.** Let us assume that $T_{Zi}(\dot{X}, \dot{G}) = Z_{\perp} \dot{X} + Z \dot{G} = 0$. Multiplying this equality by $Z^+$ and $Z_{\perp}^+$ on the left, we obtain $\dot{G} = 0$ and $\dot{X} = 0$, respectively, which implies that $T_{Zi}$ is injective. To prove that it is also surjective, we consider a matrix $\dot{Z} \in \mathbb{R}^{k \times r}$ and observe that $\dot{X} = Z_{\perp}^{+} \dot{Z} \in \mathbb{R}^{(k-r) \times r}$ and $\dot{G} = Z^{+} \dot{Z} \in \mathbb{R}^{r \times r}$ is such that $T_{Zi}(\dot{X}, \dot{G}) = \dot{Z}$. Since $T_{Zi}$ is injective, the inclusion map $i$ is an immersion.

To prove that it is an embedding, we equip $M_r(\mathbb{R}^{k \times r})$ with the topology $\tau_{\mathcal{B}_{k,r}}$ given by the atlas and we equip $\mathbb{R}^{k \times r}$ with the topology $\tau_{\mathbb{R}^{k \times r}}$ induced by matrix norms. We need to check that

$$i : (M_r(\mathbb{R}^{k \times r}), \tau_{\mathcal{B}_{k,r}}) \to (M_r(\mathbb{R}^{k \times r}), \tau_{\mathbb{R}^{k \times r}|_{M_r(\mathbb{R}^{k \times r})})$$

is a topological homeomorphism. Since the topology in $(M_r(\mathbb{R}^{k \times r}), \tau_{\mathcal{B}_{k,r}})$ has the property that each local chart $\xi_Z$ is indeed a homeomorphism from $\mathcal{V}_Z$ in $M_r(\mathbb{R}^{k \times r})$ to $\xi_Z(\mathcal{V}_Z) = \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r$ (see Section 1.1), we only need to show that the bijection $(i \circ \xi_Z^{-1}) : \mathbb{R}^{(k-r) \times r} \times \mathrm{GL}_r \to \mathcal{V}_Z \subset \mathbb{R}^{k \times r}$ given by

$$(i \circ \xi_Z^{-1})(X, G) = (Z + Z_{\perp} X)G$$

is a topological homeomorphism for all $Z \in M_r(\mathbb{R}^{k \times r})$. Observe that $D(i \circ \xi_Z^{-1})(X, G) \in \mathcal{L}(\mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r}, \mathbb{R}^{k \times r})$ is given by

$$D(i \circ \xi_Z^{-1})(X, G)[(\dot{X}, \dot{G})] = Z_{\perp} \dot{X} G + (Z + Z_{\perp} X) \dot{G}.$$

Assume that $Z_{\perp} \dot{X} G + (Z + Z_{\perp} X) \dot{G} = 0$. Multiplying this equality by $Z^+$ on the left we obtain $\dot{G} = 0$, and hence $Z_{\perp} \dot{X} G = 0$. Multiplying by $Z_{\perp}^+$ on the left, we obtain
$\dot{X}G = 0$. Thus, $\dot{X} = 0$ and as a consequence $D(i \circ \xi_Z^{-1})(X, G)$ is a linear isomorphism for each $(X, G) \in \mathbb{R}^{(k-r) \times r} \times GL_r$. The inverse function theorem says us that $(i \circ \xi_Z^{-1})$ is a diffeomorphism, in particular a homeomorphism, and hence $i$ is an embedding. $\square$

The tangent space to $\mathcal{M}_r(\mathbb{R}^{k \times r})$ at $Z$ is the image through $T_Z i$ of the tangent space at $Z$ in local coordinates $T_Z \mathcal{M}_r(\mathbb{R}^{k \times r}) = \mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r}$, i.e.,

$$T_Z \mathcal{M}_r(\mathbb{R}^{k \times r}) = \{ Z_{\perp} \dot{X} + Z \dot{G} : \dot{X} \in \mathbb{R}^{(k-r) \times r}, \dot{G} \in \mathbb{R}^{r \times r} \} = \mathbb{R}^{k \times r}, $$

and can be decomposed into a vertical tangent space

$$T_Z^V \mathcal{M}_r(\mathbb{R}^{k \times r}) = \{ Z \dot{G} : \dot{G} \in \mathbb{R}^{r \times r} \}, $$

and a horizontal tangent space

$$T_Z^H \mathcal{M}_r(\mathbb{R}^{k \times r}) = \{ Z_{\perp} \dot{X} : \dot{X} \in \mathbb{R}^{(k-r) \times r} \}. $$

### 3.3. Lie Group Structure of Neighbourhoods $V_Z$

We here prove that each neighbourhood $V_Z$ of $\mathcal{M}_r(\mathbb{R}^{k \times r})$ has the structure of a Lie group. For that, we first note that $V_Z$ can be identified with $S_Z \times GL_r$, with $S_Z$ given by (9). Noting that $S_Z$ can be identified with the Lie group $G_Z$ defined in (11), we then have that $V_Z$ can be identified with a product of two Lie groups $G_Z \times GL_r$, which is a Lie group with the group operation $\odot_Z$ given by

$$(\exp(Z_{\perp}XZ^{+}), G) \odot_Z (\exp(Z_{\perp}X'Z^{+}), G') = (\exp(Z_{\perp}(X + X')Z^{+}), GG'), $$

for $X, X' \in \mathbb{R}^{(k-r) \times r}$ and $G, G' \in GL_r$. This allows us to define a group operation $\star_Z$ over $V_Z$ defined for $W = \xi_Z^{-1}(X, G)$ and $W' = \xi_Z^{-1}(X', G')$ by

$$W \star_Z W' = \xi_Z^{-1}(X + X', GG'), \qquad (15)$$

and to state the following result.

**Theorem 6.** The set $V_Z$ together with the group operation $\star_Z$ defined by (15) is a Lie group and the map $\eta_Z : V_Z \to G_Z \times GL_r$ given by

$$\eta_Z(\xi_Z^{-1}(X, G)) = (\exp(Z_{\perp}XZ^{+}), G),$$

is a Lie group isomorphism.

## 4. The Principal Bundle $\mathcal{M}_r(\mathbb{R}^{n \times m})$ for $0 < r < \min(m, n)$

In this section, we give a geometric description of the set of matrices $\mathcal{M}_r(\mathbb{R}^{n \times m})$ with rank $r < \min(m, n)$.

### 4.1. $\mathcal{M}_r(\mathbb{R}^{n \times m})$ as a Principal Bundle

For $Z \in \mathcal{M}_r(\mathbb{R}^{n \times m})$, there exists $U \in \mathcal{M}_r(\mathbb{R}^{n \times r})$, $V \in \mathcal{M}_r(\mathbb{R}^{m \times r})$, and $G \in GL_r$ such that

$$Z = UGV^T,$$

where the column space of $Z$ is $\text{col}_{n,r}(U)$ and the row space of $Z$ is $\text{col}_{m,r}(V)$.

Let us first introduce the surjective map

$$\varrho_r : \mathcal{M}_r(\mathbb{R}^{n \times m}) \to G_r(\mathbb{R}^n) \times G_r(\mathbb{R}^m), \quad UGV^T \mapsto (\text{col}_{n,r}(U), \text{col}_{m,r}(V)).$$

The set

$$\varrho_r^{-1}(\text{col}_{n,r}(U), \text{col}_{m,r}(V)) = \{ UHV^T : H \in GL_r \}$$
$\dot{X}G = 0$. Thus, $\dot{X} = 0$ and as a consequence $D(i \circ \xi_Z^{-1})(X, G)$ is a linear isomorphism for each $(X, G) \in \mathbb{R}^{(k-r) \times r} \times GL_r$. The inverse function theorem says us that $(i \circ \xi_Z^{-1})$ is a diffeomorphism, in particular a homeomorphism, and hence $i$ is an embedding. $\square$

The tangent space to $\mathcal{M}_r(\mathbb{R}^{k \times r})$ at $Z$ is the image through $T_Z i$ of the tangent space at $Z$ in local coordinates $T_Z \mathcal{M}_r(\mathbb{R}^{k \times r}) = \mathbb{R}^{(k-r) \times r} \times \mathbb{R}^{r \times r}$, i.e.,

$$T_Z \mathcal{M}_r(\mathbb{R}^{k \times r}) = \{ Z_{\perp} \dot{X} + Z \dot{G} : \dot{X} \in \mathbb{R}^{(k-r) \times r}, \dot{G} \in \mathbb{R}^{r \times r} \} = \mathbb{R}^{k \times r}, $$

and can be decomposed into a vertical tangent space

$$T_Z^V \mathcal{M}_r(\mathbb{R}^{k \times r}) = \{ Z \dot{G} : \dot{G} \in \mathbb{R}^{r \times r} \}, $$

and a horizontal tangent space

$$T_Z^H \mathcal{M}_r(\mathbb{R}^{k \times r}) = \{ Z_{\perp} \dot{X} : \dot{X} \in \mathbb{R}^{(k-r) \times r} \}. $$

### 3.3. Lie Group Structure of Neighbourhoods $V_Z$

We here prove that each neighbourhood $V_Z$ of $\mathcal{M}_r(\mathbb{R}^{k \times r})$ has the structure of a Lie group. For that, we first note that $V_Z$ can be identified with $S_Z \times GL_r$, with $S_Z$ given by (9). Noting that $S_Z$ can be identified with the Lie group $G_Z$ defined in (11), we then have that $V_Z$ can be identified with a product of two Lie groups $G_Z \times GL_r$, which is a Lie group with the group operation $\odot_Z$ given by

$$(\exp(Z_{\perp} X Z^{+}), G) \odot_Z (\exp(Z_{\perp} X' Z^{+}), G') = (\exp(Z_{\perp}(X + X') Z^{+}), GG'), $$

for $X, X' \in \mathbb{R}^{(k-r) \times r}$ and $G, G' \in GL_r$. This allows us to define a group operation $\star_Z$ over $V_Z$ defined for $W = \xi_Z^{-1}(X, G)$ and $W' = \xi_Z^{-1}(X', G')$ by

$$W \star_Z W' = \xi_Z^{-1}(X + X', GG'), \qquad (15)$$

and to state the following result.

**Theorem 6.** The set $V_Z$ together with the group operation $\star_Z$ defined by (15) is a Lie group and the map $\eta_Z : V_Z \to G_Z \times GL_r$ given by

$$\eta_Z(\xi_Z^{-1}(X, G)) = (\exp(Z_{\perp} X Z^{+}), G),$$

is a Lie group isomorphism.

## 4. The Principal Bundle $\mathcal{M}_r(\mathbb{R}^{n \times m})$ for $0 < r < \min(m, n)$

In this section, we give a geometric description of the set of matrices $\mathcal{M}_r(\mathbb{R}^{n \times m})$ with rank $r < \min(m, n)$.

### 4.1. $\mathcal{M}_r(\mathbb{R}^{n \times m})$ as a Principal Bundle

For $Z \in \mathcal{M}_r(\mathbb{R}^{n \times m})$, there exists $U \in \mathcal{M}_r(\mathbb{R}^{n \times r})$, $V \in \mathcal{M}_r(\mathbb{R}^{m \times r})$, and $G \in GL_r$ such that

$$Z = UGV^T,$$

where the column space of Z is $\text{col}_{n,r}(U)$ and the row space of Z is $\text{col}_{m,r}(V)$.

Let us first introduce the surjective map

$$\varrho_r : \mathcal{M}_r(\mathbb{R}^{n \times m}) \to G_r(\mathbb{R}^n) \times G_r(\mathbb{R}^m), \quad UGV^T \mapsto (\text{col}_{n,r}(U), \text{col}_{m,r}(V)).$$

The set

$$\varrho_r^{-1}(\text{col}_{n,r}(U), \text{col}_{m,r}(V)) = \{ UHV^T : H \in GL_r \}$$
can be identified with GL<sub>r</sub>. Let us consider U<sub>⊥</sub> ∈ M<sub>n-r</sub>(R<sup>n×(n-r)</sup>) such that U<sup>T</sup> U<sub>⊥</sub> = 0 and V<sub>⊥</sub> ∈ M<sub>m-r</sub>(R<sup>m×(m-r)</sup>) such that V<sup>T</sup> V<sub>⊥</sub> = 0 (see Remark 1 for a practical definition). Then we define a neighbourhood of UGV<sup>T</sup> in the set M<sub>r</sub>(R<sup>n×m</sup>) by

$$
\mathcal{U}_Z := \rho_r^{-1}(\mathfrak{U}_{\mathcal{U}_I} \times \mathfrak{U}_V),
$$

where U<sub>I</sub> and U<sub>V</sub> are the neighbourhoods of col<sub>n,r</sub>(U) and col<sub>m,r</sub>(V), respectively (see Section 2.2). Noting that U<sub>I</sub> = q<sub>U</sub><sup>-1</sup>(R<sup>(n-r)×r</sup>) = col<sub>n,r</sub>(S<sub>U</sub>) and U<sub>V</sub> = q<sub>V</sub><sup>-1</sup>(R<sup>(m-r)×r</sup>) = col<sub>m,r</sub>(S<sub>V</sub>), where S<sub>U</sub> and S<sub>V</sub> are the affine cross sections of U and V, respectively (defined by (4)), the neighbourhood of UGV<sup>T</sup> can be written

$$
\mathcal{U}_Z = \{(U + U_{\perp}X)H(V + V_{\perp}Y)^T : (X, Y, H) \in \mathbb{R}^{(n-r)\times r} \times \mathbb{R}^{(m-r)\times r} \times \mathrm{GL}_r\}.
$$

We can associate with $\mathcal{U}_Z$ the parametrisation $\theta_Z^{-1}$ given by the chart (see Figure 2)

$$
\theta_Z : \mathcal{U}_Z \to \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r
$$

defined by

$$
\theta_Z^{-1}(X, Y, H) = (U + U_{\perp}X)H(V + V_{\perp}Y)^T
$$

for $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$, and

$$
\theta_Z(A) = (U_{\perp}^+ A (V^+)^T (U^+ A (V^+)^T)^{-1}, V_{\perp}^+ A^T (U^+)^T (V^+ A^T (U^+)^T)^{-1}, U^+ A (V^+)^T)
$$

for $A \in \mathcal{U}_Z$. In particular, we have $\theta_Z^{-1}(0,0,G) = Z$. We point out that $\mathcal{U}_Z = \mathcal{U}_{Z'}$ and $\theta_Z = \theta_{Z'}$ for every $Z' = UG'V^T$ with $G' \neq G$.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Illustration of the chart $\theta_Z$ which associates with $W = (U + U_{\perp}X)H(V + V_{\perp}Y)^T \in \mathcal{U}_Z \subset \mathcal{M}_r(\mathbb{R}^{n \times m})$, the parameters $(X, Y, G)$ in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$.</figcaption></figure>

**Theorem 7.** The collection $\mathcal{B}_{n,m,r} := \{(\mathcal{U}_Z, \theta_Z) : Z \in \mathcal{M}_r(\mathbb{R}^{n \times m})\}$ is an analytic atlas for $\mathcal{M}_r(\mathbb{R}^{n \times m})$, and hence $(\mathcal{M}_r(\mathbb{R}^{n \times m}), \mathcal{B}_{n,m,r})$ is an analytic $r(n+m-r)$-dimensional manifold modelled on $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}$.

**Proof.** {$\mathcal{U}_Z$}$_{Z \in \mathcal{M}_r(\mathbb{R}^{n \times m})}$ is clearly a covering of $\mathcal{M}_r(\mathbb{R}^{n \times m})$. Moreover, since $\theta_Z$ is bijective from $\mathcal{U}_Z$ to $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$, we claim that if $\mathcal{U}_Z \cap \mathcal{U}_{\bar{Z}} \neq \emptyset$ for $Z = UGV^T$ and $\bar{Z} = \tilde{U}\tilde{G}\tilde{V}^T \in \mathcal{M}_r(\mathbb{R}^{n \times m})$, then the following statements hold:

(i) $\theta_Z(\mathcal{U}_Z \cap \mathcal{U}_{\bar{Z}})$ and $\theta_{\bar{Z}}(\mathcal{U}_Z \cap \mathcal{U}_{\bar{Z}})$ are open sets in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ and

(ii) the map $\theta_{\bar{Z}} \circ \theta_Z^{-1}$ is analytic from $\theta_Z(\mathcal{U}_Z \cap \mathcal{U}_{\bar{Z}}) \subset \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ to $\theta_Z(\mathcal{U}_Z \cap \mathcal{U}_{\bar{Z}}) \subset \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$.

In this proof, we equip $\mathbb{R}^{n \times m}$ with the topology $\tau_{\mathbb{R}^{n \times m}}$ induced by matrix norms. We first observe that the set $\mathcal{U}_Z = \{A \in \mathcal{M}_r(\mathbb{R}^{n \times m}) : \det(U^T A V) \neq 0\} = \mathcal{O}_Z \cap \mathcal{M}_r(\mathbb{R}^{n \times m})$, where $\mathcal{O}_Z = \{A \in \mathbb{R}^{n \times m} : \det(U^T A V) \neq 0\}$, as the inverse image of the open set $\mathbb{R} \setminus \{0\}$ through the continuous map $A \mapsto \det(U^T A V)$ from $\mathbb{R}^{n \times m}$ to $\mathbb{R}$, is an open set in $\mathbb{R}^{n \times m}$. In the same way, we have that $\mathcal{U}_{\bar{Z}} = \mathcal{O}_{\bar{Z}} \cap \mathcal{M}_r(\mathbb{R}^{n \times m})$, with $\mathcal{U}_{\bar{Z}}$ as an open set in $\mathbb{R}^{n \times m}$.
can be identified with GL$_r$. Let us consider $U_\perp \in \mathcal{M}_{n-r}(\mathbb{R}^{n \times (n-r)})$ such that $U^T U_\perp = 0$ and $V_\perp \in \mathcal{M}_{m-r}(\mathbb{R}^{m \times (m-r)})$ such that $V^T V_\perp = 0$ (see Remark 1 for a practical definition). Then we define a neighbourhood of $UGV^T$ in the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ by

$$ U_Z := \rho_r^{-1}(\mathfrak{U}_U \times \mathfrak{U}_V), $$

where $\mathfrak{U}_U$ and $\mathfrak{U}_V$ are the neighbourhoods of $\mathrm{col}_{n,r}(U)$ and $\mathrm{col}_{m,r}(V)$, respectively (see Section 2.2). Noting that $\mathfrak{U}_U = \varphi_U^{-1}(\mathbb{R}^{(n-r) \times r}) = \mathrm{col}_{n,r}(S_U)$ and $\mathfrak{U}_V = \varphi_V^{-1}(\mathbb{R}^{(m-r) \times r}) = \mathrm{col}_{m,r}(S_V)$, where $S_U$ and $S_V$ are the affine cross sections of $U$ and $V$, respectively (defined by (4)), the neighbourhood of $UGV^T$ can be written

$$ U_Z = \{(U + U_\perp X)H(V + V_\perp Y)^T : (X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r\}. $$

We can associate with $U_Z$ the parametrisation $\theta_Z^{-1}$ given by the chart (see Figure 2)

$$ \theta_Z : U_Z \to \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r $$

defined by

$$ \theta_Z^{-1}(X, Y, H) = (U + U_\perp X)H(V + V_\perp Y)^T $$

for $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$, and

$$ \theta_Z(A) = (U_\perp^+ A (V^+)^T (U^+ A (V^+)^T)^{-1}, V_\perp^+ A^T (U^+)^T (V^+ A^T (U^+)^T)^{-1}, U^+ A (V^+)^T) $$

for $A \in U_Z$. In particular, we have $\theta_Z^{-1}(0, 0, G) = Z$. We point out that $U_Z = U_{Z'}$ and $\theta_Z = \theta_{Z'}$ for every $Z' = UG'V^T$ with $G' \neq G$.

<figure><img src="image_4.png" /><figcaption>Figure 2. Illustration of the chart $\theta_Z$ which associates with $W = (U + U_\perp X)H(V + V_\perp Y)^T \in U_Z \subset \mathcal{M}_r(\mathbb{R}^{n \times m})$, the parameters $(X, Y, G)$ in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$.</figcaption></figure>

**Theorem 7.** The collection $\mathcal{B}_{n,m,r} := \{(U_Z, \theta_Z) : Z \in \mathcal{M}_r(\mathbb{R}^{n \times m})\}$ is an analytic atlas for $\mathcal{M}_r(\mathbb{R}^{n \times m})$, and hence $(\mathcal{M}_r(\mathbb{R}^{n \times m}), \mathcal{B}_{n,m,r})$ is an analytic $r(n+m-r)$-dimensional manifold modelled on $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}$.

**Proof.** $\{U_Z\}_{Z \in \mathcal{M}_r(\mathbb{R}^{n \times m})}$ is clearly a covering of $\mathcal{M}_r(\mathbb{R}^{n \times m})$. Moreover, since $\theta_Z$ is bijective from $U_Z$ to $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$, we claim that if $U_Z \cap U_{\bar{Z}} \neq \emptyset$ for $Z = UGV^T$ and $\bar{Z} = \tilde{U}\tilde{G}\tilde{V}^T \in \mathcal{M}_r(\mathbb{R}^{n \times m})$, then the following statements hold:

(i) $\theta_Z(U_Z \cap U_{\bar{Z}})$ and $\theta_{\bar{Z}}(U_Z \cap U_{\bar{Z}})$ are open sets in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ and

(ii) the map $\theta_{\bar{Z}} \circ \theta_Z^{-1}$ is analytic from $\theta_Z(U_Z \cap U_{\bar{Z}}) \subset \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ to $\theta_Z(U_Z \cap U_{\bar{Z}}) \subset \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$.

In this proof, we equip $\mathbb{R}^{n \times m}$ with the topology $\tau_{\mathbb{R}^{n \times m}}$ induced by matrix norms. We first observe that the set $U_Z = \{A \in \mathcal{M}_r(\mathbb{R}^{n \times m}) : \det(U^T A V) \neq 0\} = \mathcal{O}_Z \cap \mathcal{M}_r(\mathbb{R}^{n \times m})$, where $\mathcal{O}_Z = \{A \in \mathbb{R}^{n \times m} : \det(U^T A V) \neq 0\}$, as the inverse image of the open set $\mathbb{R} \setminus \{0\}$ through the continuous map $A \mapsto \det(U^T A V)$ from $\mathbb{R}^{n \times m}$ to $\mathbb{R}$, is an open set in $\mathbb{R}^{n \times m}$. In the same way, we have that $U_{\bar{Z}} = \mathcal{O}_{\bar{Z}} \cap \mathcal{M}_r(\mathbb{R}^{n \times m})$, with $U_{\bar{Z}}$ as an open set in $\mathbb{R}^{n \times m}$.
Since $U_Z \cap U_{\bar{Z}} = O_Z \cap O_{\bar{Z}} \cap M_r(\mathbb{R}^{n \times m})$, and since the image of $\theta_Z^{-1}$ is in $M_r(\mathbb{R}^{n \times m})$, we have

$$ \theta_Z(U_Z \cap U_{\bar{Z}}) = (\theta_Z^{-1})^{-1}(U_Z \cap U_{\bar{Z}}) = (\theta_Z^{-1})^{-1}(O_Z \cap O_{\bar{Z}}), $$

the inverse image through $\theta_Z^{-1}$ of the open set $O_Z \cap O_{\bar{Z}}$ in $\mathbb{R}^{n \times m}$. Since $\theta_Z^{-1}$ is a continuous map from $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$ to $\mathbb{R}^{n \times m}$, we deduce that $\theta_Z(U_Z \cap U_{\bar{Z}})$ is an open set in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$. Similarly, $\theta_{\bar{Z}}(U_{\bar{Z}} \cap U_Z)$ is an open set in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$. Now, let $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$ such that $\theta_Z^{-1}(X, Y, H) \in U_Z \cap U_{\bar{Z}}$. From the expressions of $\theta_Z^{-1}$ and $\theta_{\bar{Z}}$, the map $\theta_{\bar{Z}} \circ \theta_Z^{-1}$ is defined by

$$ \theta_{\bar{Z}} \circ \theta_Z^{-1}(X, Y, H) = (\tilde{U}_{\perp}^{+} \theta_Z^{-1}(X, Y, H)(\tilde{V}^{+})^{T}(\tilde{U}^{+} \theta_Z^{-1}(X, Y, H)(\tilde{V}^{+})^{T})^{-1}, \\ \tilde{V}_{\perp}^{+} \theta_Z^{-1}(X, Y, H)^{T}(\tilde{U}^{+})^{T}(\tilde{V}^{+} \theta_Z^{-1}(X, Y, H)^{T}(\tilde{U}^{+})^{T})^{-1}, \\ \tilde{U}^{+} \theta_Z^{-1}(X, Y, H)(\tilde{V}^{+})^{T}), $$

with $\theta_Z^{-1}(X, Y, H) = (U + U_{\perp}X)H(V + V_{\perp}Y)^T$, which is clearly an analytic map. $\square$

**Theorem 8.** The set $M_r(\mathbb{R}^{n \times m})$ is an analytic principal bundle with typical fibre $\text{GL}_r$ and base $\text{G}_r(\mathbb{R}^n) \times \text{G}_r(\mathbb{R}^m)$ with surjective morphism $\varrho_r$ between $M_r(\mathbb{R}^{n \times m})$ and $\text{G}_r(\mathbb{R}^n) \times \text{G}_r(\mathbb{R}^m)$ given by $\varrho_r$.

**Proof.** To prove that it is an analytic principal bundle, we consider the surjective map

$$ \varrho_r : M_r(\mathbb{R}^{n \times m}) \to \text{G}_r(\mathbb{R}^n) \times \text{G}_r(\mathbb{R}^m), \quad UGV^T \mapsto (\text{col}_{n,r}(U), \text{col}_{m,r}(V)), $$

the atlas $A_{n,r} := \{(\mathfrak{U}_U, \varphi_U) : U \in M_r(\mathbb{R}^{n \times r})\}$ of $\text{G}_r(\mathbb{R}^n)$ and the atlas $A_{m,r} := \{(\mathfrak{U}_V, \varphi_V) : V \in M_r(\mathbb{R}^{m \times r})\}$ of $\text{G}_r(\mathbb{R}^m)$. Recall that

$$ \mathfrak{U}_Z = \{\text{col}_{k,r}(Z + Z_{\perp}X) : X \in \mathbb{R}^{(k-r) \times r}\}, $$

with $k = n$ if $Z = U$ or $k = m$ if $Z = V$, and hence

$$ \varrho_r^{-1}(\mathfrak{U}_U, \mathfrak{U}_V) = \{ (U + U_{\perp}X)H(V + V_{\perp}Y)^T : X \in \mathbb{R}^{(n-r) \times r}, Y \in \mathbb{R}^{(m-r) \times r}, H \in \text{GL}_r \}. $$

Observe that for each fixed $G \in \text{GL}_r$, we have that $\varrho_r^{-1}(\mathfrak{U}_U, \mathfrak{U}_V) = U_Z$, where $Z = UGV^T$. Since $U_Z = U_{\bar{Z}}$ holds for $Z' = UG'V^T$, where $G' \in \text{GL}_r$, the map

$$ \chi_Z : U_Z \to \mathfrak{U}_U \times \mathfrak{U}_V \times \text{GL}_r $$

defined by

$$ \chi_Z(U' H'(V')^T) := (\text{col}_{n,r}(U'), \text{col}_{m,r}(V'), H'), $$

is independent of the choice of $Z = UGV^T$, where $G \in \text{GL}_r$. Now, the representation of $\chi_Z$ in local coordinates is the map

$$ ((\varphi_U \times \varphi_V \times \text{id}_{\mathbb{R}^{r \times r}}) \circ \chi_Z \circ \theta_Z^{-1}) : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r \to \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r $$

given by $((\varphi_U \times \varphi_V \times \text{id}_{\mathbb{R}^{r \times r}}) \circ \chi_Z \circ \theta_Z^{-1})(X, Y, H) = (X, Y, H)$, which is an analytic diffeomorphism. Moreover, let $p_Z : \mathfrak{U}_U \times \mathfrak{U}_V \times \text{GL}_r \to \mathfrak{U}_U \times \mathfrak{U}_V$ be the projection over the first two components. Then

$$ (p_Z \circ \chi_Z)(UHV^T) = (\text{col}_{n,r}(U), \text{col}_{m,r}(V)) = \varrho_r(UHV^T) $$

and the theorem follows. $\square$
Since $U_Z \cap U_{\bar{Z}} = O_Z \cap O_{\bar{Z}} \cap M_r(\mathbb{R}^{n \times m})$, and since the image of $\theta_Z^{-1}$ is in $M_r(\mathbb{R}^{n \times m})$, we have

$$ \theta_Z(U_Z \cap U_{\bar{Z}}) = (\theta_Z^{-1})^{-1}(U_Z \cap U_{\bar{Z}}) = (\theta_Z^{-1})^{-1}(O_Z \cap O_{\bar{Z}}), $$

the inverse image through $\theta_Z^{-1}$ of the open set $O_Z \cap O_{\bar{Z}}$ in $\mathbb{R}^{n \times m}$. Since $\theta_Z^{-1}$ is a continuous map from $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$ to $\mathbb{R}^{n \times m}$, we deduce that $\theta_Z(U_Z \cap U_{\bar{Z}})$ is an open set in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$. Similarly, $\theta_{\bar{Z}}(U_{\bar{Z}} \cap U_Z)$ is an open set in $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$. Now, let $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$ such that $\theta_Z^{-1}(X, Y, H) \in U_Z \cap U_{\bar{Z}}$. From the expressions of $\theta_Z^{-1}$ and $\theta_{\bar{Z}}$, the map $\theta_{\bar{Z}} \circ \theta_Z^{-1}$ is defined by

$$ \theta_{\bar{Z}} \circ \theta_Z^{-1}(X, Y, H) = (\tilde{U}_{\perp}^{+} \theta_Z^{-1}(X, Y, H)(\tilde{V}^{+})^{T}(\tilde{U}^{+} \theta_Z^{-1}(X, Y, H)(\tilde{V}^{+})^{T})^{-1}, \\ \tilde{V}_{\perp}^{+} \theta_Z^{-1}(X, Y, H)^{T}(\tilde{U}^{+})^{T}(\tilde{V}^{+} \theta_Z^{-1}(X, Y, H)^{T}(\tilde{U}^{+})^{T})^{-1}, \\ \tilde{U}^{+} \theta_Z^{-1}(X, Y, H)(\tilde{V}^{+})^{T}), $$

with $\theta_Z^{-1}(X, Y, H) = (U + U_{\perp}X)H(V + V_{\perp}Y)^T$, which is clearly an analytic map. $\square$

**Theorem 8.** The set $M_r(\mathbb{R}^{n \times m})$ is an analytic principal bundle with typical fibre $\text{GL}_r$ and base $\text{G}_r(\mathbb{R}^n) \times \text{G}_r(\mathbb{R}^m)$ with surjective morphism $\varrho_r$ between $M_r(\mathbb{R}^{n \times m})$ and $\text{G}_r(\mathbb{R}^n) \times \text{G}_r(\mathbb{R}^m)$ given by $\varrho_r$.

**Proof.** To prove that it is an analytic principal bundle, we consider the surjective map

$$ \varrho_r : M_r(\mathbb{R}^{n \times m}) \to \text{G}_r(\mathbb{R}^n) \times \text{G}_r(\mathbb{R}^m), \quad UGV^T \mapsto (\text{col}_{n,r}(U), \text{col}_{m,r}(V)), $$

the atlas $A_{n,r} := \{(\mathfrak{U}_U, \varphi_U) : U \in M_r(\mathbb{R}^{n \times r})\}$ of $\text{G}_r(\mathbb{R}^n)$ and the atlas $A_{m,r} := \{(\mathfrak{U}_V, \varphi_V) : V \in M_r(\mathbb{R}^{m \times r})\}$ of $\text{G}_r(\mathbb{R}^m)$. Recall that

$$ \mathfrak{U}_Z = \{\text{col}_{k,r}(Z + Z_{\perp}X) : X \in \mathbb{R}^{(k-r) \times r}\}, $$

with $k = n$ if $Z = U$ or $k = m$ if $Z = V$, and hence

$$ \varrho_r^{-1}(\mathfrak{U}_U, \mathfrak{U}_V) = \{ (U + U_{\perp}X)H(V + V_{\perp}Y)^T : X \in \mathbb{R}^{(n-r) \times r}, Y \in \mathbb{R}^{(m-r) \times r}, H \in \text{GL}_r \}. $$

Observe that for each fixed $G \in \text{GL}_r$, we have that $\varrho_r^{-1}(\mathfrak{U}_U, \mathfrak{U}_V) = U_Z$, where $Z = UGV^T$. Since $U_Z = U_{\bar{Z}}$ holds for $Z' = UG'V^T$, where $G' \in \text{GL}_r$, the map

$$ \chi_Z : U_Z \to \mathfrak{U}_U \times \mathfrak{U}_V \times \text{GL}_r $$

defined by

$$ \chi_Z(U' H'(V')^T) := (\text{col}_{n,r}(U'), \text{col}_{m,r}(V'), H'), $$

is independent of the choice of $Z = UGV^T$, where $G \in \text{GL}_r$. Now, the representation of $\chi_Z$ in local coordinates is the map

$$ ((\varphi_U \times \varphi_V \times \text{id}_{\mathbb{R}^{r \times r}}) \circ \chi_Z \circ \theta_Z^{-1}) : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r \to \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r $$

given by $((\varphi_U \times \varphi_V \times \text{id}_{\mathbb{R}^{r \times r}}) \circ \chi_Z \circ \theta_Z^{-1})(X, Y, H) = (X, Y, H)$, which is an analytic diffeomorphism. Moreover, let $p_Z : \mathfrak{U}_U \times \mathfrak{U}_V \times \text{GL}_r \to \mathfrak{U}_U \times \mathfrak{U}_V$ be the projection over the first two components. Then

$$ (p_Z \circ \chi_Z)(UHV^T) = (\text{col}_{n,r}(U), \text{col}_{m,r}(V)) = \varrho_r(UHV^T) $$

and the theorem follows. $\square$
### 4.2. $\mathcal{M}_r(\mathbb{R}^{n \times m})$ as a Submanifold and Its Tangent Space

Here, we prove that $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology given by the atlas $\mathcal{B}_{n,m,r}$ is an embedded submanifold in $\mathbb{R}^{n \times m}$. For that, we have to prove that the standard inclusion map $i: \mathcal{M}_r(\mathbb{R}^{n \times m}) \to \mathbb{R}^{n \times m}$ is an embedding. Noting that the inclusion map restricted to the neighbourhood $U_Z$ of $Z = UGV^T$ is identified with

$$ (i \circ \theta_Z^{-1}) : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r \to \mathbb{R}^{n \times m}, \quad (X, Y, H) \mapsto (U + U_{\perp}X)H(V + V_{\perp}Y)^T, $$

the tangent map $T_{Zi}$ at $Z = \theta_Z^{-1}(0, 0, G)$, defined by $T_{Zi} = D(i \circ \theta_Z^{-1})(0, 0, G)$, is

$$ T_{Zi} : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{n \times m}, \quad (\dot{X}, \dot{Y}, \dot{H}) \mapsto U_{\perp} \dot{X} GV^T + UG(V_{\perp} \dot{Y})^T + U \dot{H} V^T. $$

**Proposition 6.** The tangent map $T_{Zi}: \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{n \times m}$ at $Z = UGV^T \in \mathcal{M}_r(\mathbb{R}^{n \times m})$ is a linear isomorphism with inverse $(T_{Zi})^{-1}$ given by

$$ (T_{Zi})^{-1}(\dot{Z}) = (U_{\perp}^{+} \dot{Z}(V^{+})^{T} G^{-1}, V_{\perp}^{+} \dot{Z}^{T}(U^{+})^{T} G^{-T}, U^{+} \dot{Z}(V^{+})^{T}) $$

for $\dot{Z} \in \mathbb{R}^{n \times m}$. Furthermore, the standard inclusion map $i$ is an embedding from $\mathcal{M}_r(\mathbb{R}^{n \times m})$ to $\mathbb{R}^{n \times m}$.

**Proof.** Let us suppose that $T_{Zi}(\dot{X}, \dot{Y}, \dot{H}) = 0$. Multiplying this equality by $(U_{\perp})^+$ and $U^+$ on the left leads to

$$ \dot{X}GV^T = 0 \text{ and } G(V_{\perp} \dot{Y})^T + \dot{H}V^T = 0, $$

respectively. By multiplying the first equation by $(V^+)^T$ on the right, we obtain $\dot{X} = 0$. By multiplying the second equation on the right by $(V^+)^T$ and $(V_{\perp}^+)^T$, we respectively obtain $\dot{H} = 0$ and $\dot{Y} = 0$. Then, $T_{Zi}$ is injective and then $i$ is an immersion. For $\dot{Z} \in \mathbb{R}^{n \times m}$, we note that $\dot{X} = U_{\perp}^{+} \dot{Z}(V^+)^T G^{-1} \in \mathbb{R}^{n \times r}$, $\dot{Y} = V_{\perp}^{+} \dot{Z}^T(U^+)^T G^{-T} \in \mathbb{R}^{m \times r}$, and $\dot{G} = U^+ \dot{Z}(V^+)^T \in \mathbb{R}^{r \times r}$ is such that $T_{Zi}(\dot{X}, \dot{Y}, \dot{G}) = \dot{Z}$, and $T_{Zi}$ is also surjective. Let us now equip $\mathcal{M}_r(\mathbb{R}^{n \times m})$ with the topology $\tau_{\mathcal{B}_{n,m,r}}$ given by the atlas and $\mathbb{R}^{n \times m}$ with the topology $\tau_{\mathbb{R}^{n \times m}}$ induced by matrix norms. We have to prove that

$$ i: (\mathcal{M}_r(\mathbb{R}^{n \times m}), \tau_{\mathcal{B}_{n,m,r}}) \to (\mathcal{M}_r(\mathbb{R}^{n \times m}), \tau_{\mathbb{R}^{n \times m}|\mathcal{M}_r(\mathbb{R}^{n \times m})}) $$

is a topological isomorphism. The topology in $(\mathcal{M}_r(\mathbb{R}^{n \times m}), \tau_{\mathcal{B}_{n,m,r}})$ is such that a local chart $\theta_Z$ is a homeomorphism from $U_Z \subset \mathcal{M}_r(\mathbb{R}^{n \times m})$ to $\theta_Z(U_Z) = \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ (see Section 1.1). Then, to prove that the map $i$ is an embedding, we need to show that the bijection

$$ (i \circ \theta_Z^{-1}) : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r \to U_Z \subset \mathbb{R}^{n \times m} $$

is a topological homeomorphism. For that, observe that its differential

$$ D(i \circ \theta_Z^{-1})(X, Y, H) \in \mathcal{L}(\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}, \mathbb{R}^{n \times m}) $$

at $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ is given by

$$ \begin{aligned} D(i \circ \theta_Z^{-1})(X, Y, H)[(\dot{X}, \dot{Y}, \dot{H})] \\ = (U_{\perp} \dot{X}) H(V + V_{\perp} Y)^T + (U + U_{\perp} X) H(V_{\perp} \dot{Y})^T + (U + U_{\perp} X) \dot{H}(V + V_{\perp} Y)^T. \end{aligned} $$

Assume that

$$ (U_{\perp} \dot{X}) H(V + V_{\perp} Y)^T + (U + U_{\perp} X) H(V_{\perp} \dot{Y})^T + (U + U_{\perp} X) \dot{H}(V + V_{\perp} Y)^T = 0. \quad (16) $$

Multiplying on the left by $U^+$ and on the right by $(V^+)^T$, we obtain $\dot{H} = 0$. Multiplying on the left by $U_{\perp}^+$ and on the right by $(V^+)^T$ we deduce that $\dot{X}H = 0$, that is, $\dot{X} = 0$. Finally, multiplying on the left by $U^+$ and on the right by $(V_{\perp}^+)^T$, we obtain
### 4.2. $\mathcal{M}_r(\mathbb{R}^{n \times m})$ as a Submanifold and Its Tangent Space

Here, we prove that $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology given by the atlas $\mathcal{B}_{n,m,r}$ is an embedded submanifold in $\mathbb{R}^{n \times m}$. For that, we have to prove that the standard inclusion map $i: \mathcal{M}_r(\mathbb{R}^{n \times m}) \to \mathbb{R}^{n \times m}$ is an embedding. Noting that the inclusion map restricted to the neighbourhood $U_Z$ of $Z = UGV^T$ is identified with

$$ (i \circ \theta_Z^{-1}) : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r \to \mathbb{R}^{n \times m}, \quad (X, Y, H) \mapsto (U + U_{\perp} X) H (V + V_{\perp} Y)^T, $$

the tangent map $T_{Zi}$ at $Z = \theta_Z^{-1}(0, 0, G)$, defined by $T_{Zi} = D(i \circ \theta_Z^{-1})(0, 0, G)$, is

$$ T_{Zi} : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{n \times m}, \quad (\dot{X}, \dot{Y}, \dot{H}) \mapsto U_{\perp} \dot{X} GV^T + UG(V_{\perp} \dot{Y})^T + U \dot{H} V^T. $$

**Proposition 6.** The tangent map $T_{Zi}: \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r} \to \mathbb{R}^{n \times m}$ at $Z = UGV^T \in \mathcal{M}_r(\mathbb{R}^{n \times m})$ is a linear isomorphism with inverse $(T_{Zi})^{-1}$ given by

$$ (T_{Zi})^{-1}(\dot{Z}) = (U_{\perp}^{+} \dot{Z}(V^{+})^{T} G^{-1}, V_{\perp}^{+} \dot{Z}^{T}(U^{+})^{T} G^{-T}, U^{+} \dot{Z}(V^{+})^{T}) $$

for $\dot{Z} \in \mathbb{R}^{n \times m}$. Furthermore, the standard inclusion map $i$ is an embedding from $\mathcal{M}_r(\mathbb{R}^{n \times m})$ to $\mathbb{R}^{n \times m}$.

**Proof.** Let us suppose that $T_{Zi}(\dot{X}, \dot{Y}, \dot{H}) = 0$. Multiplying this equality by $(U_{\perp})^{+}$ and $U^{+}$ on the left leads to

$$ \dot{X}GV^T = 0 \text{ and } G(V_{\perp} \dot{Y})^T + \dot{H}V^T = 0, $$

respectively. By multiplying the first equation by $(V^{+})^T$ on the right, we obtain $\dot{X} = 0$. By multiplying the second equation on the right by $(V^{+})^T$ and $(V_{\perp}^{+})^T$, we respectively obtain $\dot{H} = 0$ and $\dot{Y} = 0$. Then, $T_{Zi}$ is injective and then $i$ is an immersion. For $\dot{Z} \in \mathbb{R}^{n \times m}$, we note that $\dot{X} = U_{\perp}^{+} \dot{Z}(V^{+})^T G^{-1} \in \mathbb{R}^{n \times r}$, $\dot{Y} = V_{\perp}^{+} \dot{Z}^{T}(U^{+})^T G^{-T} \in \mathbb{R}^{m \times r}$, and $\dot{G} = U^{+} \dot{Z}(V^{+})^T \in \mathbb{R}^{r \times r}$ is such that $T_{Zi}(\dot{X}, \dot{Y}, \dot{G}) = \dot{Z}$, and $T_{Zi}$ is also surjective. Let us now equip $\mathcal{M}_r(\mathbb{R}^{n \times m})$ with the topology $\tau_{\mathcal{B}_{n,m,r}}$ given by the atlas and $\mathbb{R}^{n \times m}$ with the topology $\tau_{\mathbb{R}^{n \times m}}$ induced by matrix norms. We have to prove that

$$ i: (\mathcal{M}_r(\mathbb{R}^{n \times m}), \tau_{\mathcal{B}_{n,m,r}}) \to (\mathcal{M}_r(\mathbb{R}^{n \times m}), \tau_{\mathbb{R}^{n \times m}|\mathcal{M}_r(\mathbb{R}^{n \times m})}) $$

is a topological isomorphism. The topology in $(\mathcal{M}_r(\mathbb{R}^{n \times m}), \tau_{\mathcal{B}_{n,m,r}})$ is such that a local chart $\theta_Z$ is a homeomorphism from $U_Z \subset \mathcal{M}_r(\mathbb{R}^{n \times m})$ to $\theta_Z(U_Z) = \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ (see Section 1.1). Then, to prove that the map $i$ is an embedding, we need to show that the bijection

$$ (i \circ \theta_Z^{-1}) : \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r \to U_Z \subset \mathbb{R}^{n \times m} $$

is a topological homeomorphism. For that, observe that its differential

$$ D(i \circ \theta_Z^{-1})(X, Y, H) \in \mathcal{L}(\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}, \mathbb{R}^{n \times m}) $$

at $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ is given by

$$ \begin{aligned} D(i \circ \theta_Z^{-1})(X, Y, H)[(\dot{X}, \dot{Y}, \dot{H})] \\ = (U_{\perp} \dot{X}) H(V + V_{\perp} Y)^T + (U + U_{\perp} X) H(V_{\perp} \dot{Y})^T + (U + U_{\perp} X) \dot{H}(V + V_{\perp} Y)^T. \end{aligned} $$

Assume that

$$ (U_{\perp} \dot{X}) H(V + V_{\perp} Y)^T + (U + U_{\perp} X) H(V_{\perp} \dot{Y})^T + (U + U_{\perp} X) \dot{H}(V + V_{\perp} Y)^T = 0. \quad (16) $$

Multiplying on the left by $U^{+}$ and on the right by $(V^{+})^T$, we obtain $\dot{H} = 0$. Multiplying on the left by $U_{\perp}^{+}$ and on the right by $(V^{+})^T$ we deduce that $\dot{X}H = 0$, that is, $\dot{X} = 0$. Finally, multiplying on the left by $U^{+}$ and on the right by $(V_{\perp}^{+})^T$, we obtain
tain $H\dot{Y}^T = 0$, and hence $\dot{Y} = 0$. Thus, $D(i \circ \theta_Z^{-1})(X, Y, H)$ is a linear isomorphism from $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}$ to $D(i \circ \theta_Z^{-1})(X, Y, H)[\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}]$ for each $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$. The inverse function theorem tells us that $(i \circ \theta_Z^{-1})$ is a diffeomorphism from $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r$ to $\mathcal{U}_Z = (i \circ \theta_Z^{-1})(\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \text{GL}_r)$ and, in particular, that it is a topological homeomorphism. In consequence, the map $i$ is an embedding. $\square$

The tangent space to $M_r(\mathbb{R}^{n \times m})$ at $Z = UGV^T$, which is the image through $T_Z i$ of the tangent space in local coordinates $T_Z M_r(\mathbb{R}^{n \times m}) = \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}$, is

$$T_Z M_r(\mathbb{R}^{n \times m}) = \{U_{\perp} \dot{X} G V^T + U G(V_{\perp} \dot{Y})^T + U \dot{G} V^T : \dot{X} \in \mathbb{R}^{(n-r) \times r}, \dot{Y} \in \mathbb{R}^{(m-r) \times r}, \dot{G} \in \mathbb{R}^{r \times r}\},$$

and can be decomposed into a vertical tangent space

$$T_Z^V M_r(\mathbb{R}^{n \times m}) = \{U \dot{G} V^T : \dot{G} \in \mathbb{R}^{r \times r}\},$$

and a horizontal tangent space

$$T_Z^H M_r(\mathbb{R}^{n \times m}) = \{U_{\perp} \dot{X} G V^T + U G(V_{\perp} \dot{Y})^T : \dot{X} \in \mathbb{R}^{(n-r) \times r}, \dot{Y} \in \mathbb{R}^{(m-r) \times r}\}.$$

### 4.3. Lie Group Structure of Neighbourhoods $\mathcal{U}_Z$

We here prove that $M_r(\mathbb{R}^{n \times m})$ locally has the structure of a Lie group by proving that the neighbourhoods $\mathcal{U}_Z$ can be identified with Lie groups.

Let $Z = UGV^T \in M_r(\mathbb{R}^{n \times m})$. We first note that $\mathcal{U}_Z$ can be identified with $S_U \times S_V \times \text{GL}_r$, with $S_U$ and $S_V$ defined by (9). Noting that $S_U$ and $S_V$ can be identified with Lie groups $G_U$ and $G_V$ defined in (11), we then have that $\mathcal{U}_Z$ can be identified with a product of three Lie groups, which is a Lie group with the group operation $\circ_Z$ given by

$$(\exp(U_{\perp}XU^{+}), \exp(V_{\perp}YV^{+}), G) \circ_Z (\exp(U_{\perp}X'U^{+}), \exp(V_{\perp}Y'V^{+}), G') \\ = (\exp(U_{\perp}(X + X')U^{+}), \exp(V_{\perp}(Y + Y')V^{+}), GG').$$

This allows us to define a group operation $\star_Z$ over $\mathcal{U}_Z$ defined for $W = \theta_Z^{-1}(X, Y, G)$ and $W' = \theta_Z^{-1}(X', Y', G')$ by

$$W \star_Z W' = \theta_Z^{-1}(X + X', Y + Y', GG'), \qquad (17)$$

and to state the following result.

**Theorem 9.** Let $Z = UGV^T \in M_r(\mathbb{R}^{n \times m})$. Then the set $\mathcal{U}_Z$ together with the group operation $\star_Z$ defined by (17) is a Lie group with identity element $UV^T$, and the map $\eta_Z : \mathcal{U}_Z \to G_U \times G_V \times \text{GL}_r$ given by

$$\eta_Z(\theta_Z^{-1}(X, Y, H)) = (\exp(U_{\perp}XU^{+}), \exp(V_{\perp}YV^{+}), H)$$

is a Lie group isomorphism.

**Author Contributions:** M.B.-F., A.F. and A.N. equally contributed. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the RTI2018-093521-B-C32 grant from the Ministerio de Ciencia, Innovación y Universidades and by the grant number INDI20/13 from Universidad CEU Cardenal Herrera.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
tain $H\dot{Y}^T = 0$, and hence $\dot{Y} = 0$. Thus, $D(i \circ \theta_Z^{-1})(X, Y, H)$ is a linear isomorphism from $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}$ to $D(i \circ \theta_Z^{-1})(X, Y, H)[\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}]$ for each $(X, Y, H) \in \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$. The inverse function theorem tells us that $(i \circ \theta_Z^{-1})$ is a diffeomorphism from $\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r$ to $\mathcal{U}_Z = (i \circ \theta_Z^{-1})(\mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathrm{GL}_r)$ and, in particular, that it is a topological homeomorphism. In consequence, the map $i$ is an embedding. $\square$

The tangent space to $M_r(\mathbb{R}^{n \times m})$ at $Z = UGV^T$, which is the image through $T_Z i$ of the tangent space in local coordinates $T_Z M_r(\mathbb{R}^{n \times m}) = \mathbb{R}^{(n-r) \times r} \times \mathbb{R}^{(m-r) \times r} \times \mathbb{R}^{r \times r}$, is

$$T_Z M_r(\mathbb{R}^{n \times m}) = \{U_{\perp} \dot{X} G V^T + U G(V_{\perp} \dot{Y})^T + U \dot{G} V^T : \dot{X} \in \mathbb{R}^{(n-r) \times r}, \dot{Y} \in \mathbb{R}^{(m-r) \times r}, \dot{G} \in \mathbb{R}^{r \times r}\},$$

and can be decomposed into a vertical tangent space

$$T_Z^V M_r(\mathbb{R}^{n \times m}) = \{U \dot{G} V^T : \dot{G} \in \mathbb{R}^{r \times r}\},$$

and a horizontal tangent space

$$T_Z^H M_r(\mathbb{R}^{n \times m}) = \{U_{\perp} \dot{X} G V^T + U G(V_{\perp} \dot{Y})^T : \dot{X} \in \mathbb{R}^{(n-r) \times r}, \dot{Y} \in \mathbb{R}^{(m-r) \times r}\}.$$

### 4.3. Lie Group Structure of Neighbourhoods $\mathcal{U}_Z$

We here prove that $M_r(\mathbb{R}^{n \times m})$ locally has the structure of a Lie group by proving that the neighbourhoods $\mathcal{U}_Z$ can be identified with Lie groups.

Let $Z = UGV^T \in M_r(\mathbb{R}^{n \times m})$. We first note that $\mathcal{U}_Z$ can be identified with $S_U \times S_V \times \mathrm{GL}_r$, with $S_U$ and $S_V$ defined by (9). Noting that $S_U$ and $S_V$ can be identified with Lie groups $G_U$ and $G_V$ defined in (11), we then have that $\mathcal{U}_Z$ can be identified with a product of three Lie groups, which is a Lie group with the group operation $\odot_Z$ given by

$$(\exp(U_{\perp}XU^{+}), \exp(V_{\perp}YV^{+}), G) \odot_Z (\exp(U_{\perp}X'U^{+}), \exp(V_{\perp}Y'V^{+}), G') \\ = (\exp(U_{\perp}(X + X')U^{+}), \exp(V_{\perp}(Y + Y')V^{+}), GG').$$

This allows us to define a group operation $\star_Z$ over $\mathcal{U}_Z$ defined for $W = \theta_Z^{-1}(X, Y, G)$ and $W' = \theta_Z^{-1}(X', Y', G')$ by

$$W \star_Z W' = \theta_Z^{-1}(X + X', Y + Y', GG'), \qquad (17)$$

and to state the following result.

**Theorem 9.** Let $Z = UGV^T \in M_r(\mathbb{R}^{n \times m})$. Then the set $\mathcal{U}_Z$ together with the group operation $\star_Z$ defined by (17) is a Lie group with identity element $UV^T$, and the map $\eta_Z : \mathcal{U}_Z \to G_U \times G_V \times \mathrm{GL}_r$ given by

$$\eta_Z(\theta_Z^{-1}(X, Y, H)) = (\exp(U_{\perp}XU^{+}), \exp(V_{\perp}YV^{+}), H)$$

is a Lie group isomorphism.

**Author Contributions:** M.B.-F., A.F. and A.N. equally contributed. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the RTI2018-093521-B-C32 grant from the Ministerio de Ciencia, Innovación y Universidades and by the grant number INDI20/13 from Universidad CEU Cardenal Herrera.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
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4. Nouy, A. Low-rank tensor methods for model order reduction. In *Handbook of Uncertainty Quantification*; Ghanem, R., Higdon, D., Owhadi, H., Eds.; Springer International Publishing: Cham, Switzerland, 2016.

5. Benner, P.; Cohen, A.; Ohlberger, M.; Willcox, K. (Eds.) *Model Reduction and Approximation: Theory and Algorithms*; SIAM: Philadelphia, PA, USA, 2017.

6. Kushner, A.; Lychagin, V.; Rubtsov, V. *Contact Geometry and Non-Linear Differential Equations*; Encyclopedia of Mathematics and Its Applications 101; Cambridge University Press: Cambridge, UK, 2007.

7. Goursat, E. Sur les équations du second ordre à n variables analogues à l'équation de Monge-Ampère. *Bull. Soc. Math. Fr.* **1899**, 27, 1–34. [CrossRef]

8. Sylvester, J. On the dimension of spaces of linear transformations satisfying rank conditions. *Linear Algebra Its Appl.* **1986**, *78*, 1–10. [CrossRef]

9. Eisenbud, D.; Harris, J. Vector spaces of matrices of low rank. *Adv. Math.* **1988**, *70*, 135–155. [CrossRef]

10. Westwick, R. Spaces of matrices of fixed rank. *Linear Multilinear Algebra* **1987**, *20*, 171–174. [CrossRef]

11. Westwick, R. Spaces of matrices of fixed rank II. *Linear Algebra Its Appl.* **1996**, *235*, 163–169. [CrossRef]

12. Ellia, P.; Menegatti, P. Spaces of matrices of constant rank and uniform vector bundles. *Linear Algebra Its Appl.* **2016**, *507*, 474–485. [CrossRef]

13. Smith, S.T. Optimization techniques on Riemannian manifolds. *Fields Inst. Commun.* **1994**, *3*, 113–135.

14. Absil, P.-A.; Mahony, R.; Sepulchre, R. *Optimization Algorithms on Matrix Manifolds*; Princeton University Press: Princeton, NJ, USA, 2008.

15. Vandereycken, B. Low-Rank Matrix Complet. Riemannian Optim. *SIAM J. Optim.* **2013**, *23*, 1214–1236. [CrossRef]

16. Mishra, B.; Meyer, G.; Bach, F.; Sepulchre, R. Low-rank optimization with trace norm penalty. *SIAM J. Optim.* **2013**, *23*, 2124–2149. [CrossRef]

17. Mishra, B.; Meyer, G.; Bonnabel, S.; Sepulchre, R. Fixed-rank matrix factorizations and Riemannian low-rank optimization. *Comput. Stat.* **2014**, *29*, 591–621. [CrossRef]

18. Koch, O.; Lubich, C. Dynamical low-rank approximation. *SIAM J. Matrix Anal. Appl.* **2007**, *29*, 434–454. [CrossRef]

19. Manton, J.H. A framework for generalising the Newton method and other iterative methods from Euclidean Space to manifolds. *Numer. Math.* **2015**, *129*, 91–125. [CrossRef]

20. Billaud-Friess, M.; Falcó, A.; Nouy, A. A New Splitting Algorithm for Dynamical Low-Rank Approximation Motivated by the Fibre Bundle Structure of Matrix Manifolds. *BIT Numerical Mathematics*, Accepted. Available online: https://arxiv.org/pdf/2001.08599 (accessed on 11 July 2021).

21. Michor, P.W. Gauge theory for the diffeomorphism group. In *Proceedings of the Conference Differential Geometric Methods in Theoretical Physics, Como, Italy, 24–29 August 1987*; Bleuler, K., Werner, M., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1987; pp. 345–371.

22. Falcó, A.; Hackbusch, A.W.; Nouy, A. On the Dirac-Frenkel variational principle on tensor Banach spaces. *Found. Comput. Math.* **2019**, *19*, 159–204. [CrossRef]

23. Lang, S. *Differential and Riemannian Manifolds*, 3rd ed.; Graduate Texts in Mathematics; Springer: New York, NY, USA, 1995; Volume 160.

24. Absil, P.-A.; Mahony, R.; Sepulchre, R. Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation. *Acta Appl. Math.* **2004**, *80*, 199–220. [CrossRef]

25. Hoog, F.R.D.; Mattheij, R.M.M. Subset selection for matrices. *Linear Algebra Its Appl.* **2007**, *422*, 349–359. [CrossRef]

26. Goreinov, S.A.; Oseledets, I.V.; Savostyanov, D.V.; Tyrtyshnikov, E.E.; Zamarashkin, N.L. How to find a good submatrix. In *Matrix Methods: Theory, Algorithms, Applications*; World Scientific: Hackensack, NY, USA, 2010; pp. 247–256.

27. Procesi, C. *Lie Groups: An Approach through Invariants and Representations*, 1st ed.; Springer: New York, NY, USA, 2007.

28. Abraham, R.; Marsden, J.E.; Ratiu, T. *Manifolds, Tensor Analysis, and Applications*, 2nd ed.; Springer: New York, NY, USA, 1988; Volume 75.
## References

1. Zhou, X.; Yang, C.; Zhao, H.; Yu, W. Low-Rank Modeling and Its Applications in Image Analysis. *ACM Comput. Surv.* **2014**, *47*, 1–33. [CrossRef]

2. Antoulas, A.; Sorensen, D.; Gugercin, S. A survey of model reduction methods for large-scale systems. *Contemp. Math.* **2001**, *280*, 193–220.

3. Benner, P.; Gugercin, S.; Willcox, K. A survey of projection-based model reduction methods for parametric dynamical systems. *SIAM Rev.* **2015**, *57*, 483–531. [CrossRef]

4. Nouy, A. Low-rank tensor methods for model order reduction. In *Handbook of Uncertainty Quantification*; Ghanem, R., Higdon, D., Owhadi, H., Eds.; Springer International Publishing: Cham, Switzerland, 2016.

5. Benner, P.; Cohen, A.; Ohlberger, M.; Willcox, K. (Eds.) *Model Reduction and Approximation: Theory and Algorithms*; SIAM: Philadelphia, PA, USA, 2017.

6. Kushner, A.; Lychagin, V.; Rubtsov, V. *Contact Geometry and Non-Linear Differential Equations*; Encyclopedia of Mathematics and Its Applications 101; Cambridge University Press: Cambridge, UK, 2007.

7. Goursat, E. Sur les équations du second ordre à n variables analogues à l'équation de Monge-Ampère. *Bull. Soc. Math. Fr.* **1899**, 27, 1–34. [CrossRef]

8. Sylvester, J. On the dimension of spaces of linear transformations satisfying rank conditions. *Linear Algebra Its Appl.* **1986**, *78*, 1–10. [CrossRef]

9. Eisenbud, D.; Harris, J. Vector spaces of matrices of low rank. *Adv. Math.* **1988**, *70*, 135–155. [CrossRef]

10. Westwick, R. Spaces of matrices of fixed rank. *Linear Multilinear Algebra* **1987**, *20*, 171–174. [CrossRef]

11. Westwick, R. Spaces of matrices of fixed rank II. *Linear Algebra Its Appl.* **1996**, *235*, 163–169. [CrossRef]

12. Ellia, P.; Menegatti, P. Spaces of matrices of constant rank and uniform vector bundles. *Linear Algebra Its Appl.* **2016**, *507*, 474–485. [CrossRef]

13. Smith, S.T. Optimization techniques on Riemannian manifolds. *Fields Inst. Commun.* **1994**, *3*, 113–135.

14. Absil, P.-A.; Mahony, R.; Sepulchre, R. *Optimization Algorithms on Matrix Manifolds*; Princeton University Press: Princeton, NJ, USA, 2008.

15. Vandereycken, B. Low-Rank Matrix Complet. Riemannian Optim. *SIAM J. Optim.* **2013**, *23*, 1214–1236. [CrossRef]

16. Mishra, B.; Meyer, G.; Bach, F.; Sepulchre, R. Low-rank optimization with trace norm penalty. *SIAM J. Optim.* **2013**, *23*, 2124–2149. [CrossRef]

17. Mishra, B.; Meyer, G.; Bonnabel, S.; Sepulchre, R. Fixed-rank matrix factorizations and Riemannian low-rank optimization. *Comput. Stat.* **2014**, *29*, 591–621. [CrossRef]

18. Koch, O.; Lubich, C. Dynamical low-rank approximation. *SIAM J. Matrix Anal. Appl.* **2007**, *29*, 434–454. [CrossRef]

19. Manton, J.H. A framework for generalising the Newton method and other iterative methods from Euclidean Space to manifolds. *Numer. Math.* **2015**, *129*, 91–125. [CrossRef]

20. Billaud-Friess, M.; Falcó, A.; Nouy, A. A New Splitting Algorithm for Dynamical Low-Rank Approximation Motivated by the Fibre Bundle Structure of Matrix Manifolds. *BIT Numerical Mathematics*, Accepted. Available online: https://arxiv.org/pdf/2001.08599 (accessed on 11 July 2021).

21. Michor, P.W. Gauge theory for the diffeomorphism group. In *Proceedings of the Conference Differential Geometric Methods in Theoretical Physics, Como, Italy, 24–29 August 1987*; Bleuler, K., Werner, M., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1987; pp. 345–371.

22. Falcó, A.; Hackbusch, A.W.; Nouy, A. On the Dirac-Frenkel variational principle on tensor Banach spaces. *Found. Comput. Math.* **2019**, *19*, 159–204. [CrossRef]

23. Lang, S. *Differential and Riemannian Manifolds*, 3rd ed.; Graduate Texts in Mathematics; Springer: New York, NY, USA, 1995; Volume 160.

24. Absil, P.-A.; Mahony, R.; Sepulchre, R. Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation. *Acta Appl. Math.* **2004**, *80*, 199–220. [CrossRef]

25. Hoog, F.R.D.; Mattheij, R.M.M. Subset selection for matrices. *Linear Algebra Its Appl.* **2007**, *422*, 349–359. [CrossRef]

26. Goreinov, S.A.; Oseledets, I.V.; Savostyanov, D.V.; Tyrtyshnikov, E.E.; Zamarashkin, N.L. How to find a good submatrix. In *Matrix Methods: Theory, Algorithms, Applications*; World Scientific: Hackensack, NY, USA, 2010; pp. 247–256.

27. Procesi, C. *Lie Groups: An Approach through Invariants and Representations*, 1st ed.; Springer: New York, NY, USA, 2007.

28. Abraham, R.; Marsden, J.E.; Ratiu, T. *Manifolds, Tensor Analysis, and Applications*, 2nd ed.; Springer: New York, NY, USA, 1988; Volume 75.