Principal bundle structure of matrix manifolds

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 modelled on $\mathbb{R}^{(k-r)\times r}$. Then we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ 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 $\mathrm{GL}_r$, the general linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ 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 $\mathrm{GL}_r$. The atlas of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ 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 $\mathcal{M}_r(\mathbb{R}^{n \times m})$ 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 $\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

A usual geometric approach is to endow the set M r (R n×m ) with the structure of a Riemannian manifold [16,3], which is seen as an embedded submanifold of R n×m equipped with the topology τ R n×m given by matrix norms. Standard algorithms then work in the ambient matrix space R n×m and do not rely on an explicit geometric description of the manifold using local charts (see, e.g., [17,12,13,8]). However, the matrix rank considered as a map is not continuous for the topology τ R n×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 which relies on the natural parametrization of matrices in M r (R n×m ) given by where U ∈ R n×r and V ∈ R m×r are matrices with full rank r < min{n, m}, and G ∈ R r×r is a non singular matrix. The set M r (R n×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 R n×m as an analytic manifold with a topology induced by local charts which is different from τ R n×m and for which the rank is a continuous map. Note that the representation (1) of a matrix Z is not unique because Z = (U P )(P −1 GP T )(V P −1 ) T holds for every invertible matrix P in R r×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 [8]), 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 (R n×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 or dynamical low-rank approximation, with algorithms working in local coordinates and avoiding the use of a Riemannian structure, such as in [10], where a framework is introduced for generalising iterative methods from Euclidean space to manifolds which ensures that local convergence rates are preserved. 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 [11].
We would point out 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 [7], in a tensor Banach space framework.
Before introducing the main results and outline of the paper, we recall some elements of geometry.

Elements of geometry
In this paper, we follow the approach of Serge Lang [9] 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 α ⊂ M, with α in a set A, such that {U α } α∈A is a covering of M. The next step is the explicit construction for any α ∈ A of a local chart ϕ α which is a bijection from U α to an open set X α of the finite dimensional space R Nα such that for any pair α, α ∈ M such that U α ∩ U α = ∅, the following properties hold: are open sets in X α and X α respectively, and Under the above assumptions, the set A := {(U α , ϕ α ) : α ∈ A} is an atlas which endows M with a structure of C p manifold. Then we say that (M, A) is a C p manifold, or an analytic manifold when p = ω. A consequence of the condition (ii) is that when U α ∩U α = ∅ holds for α, α ∈ A, then N α = N α . In the particular case where N α = N for all α ∈ A, we say that (M, A) is a C p manifold modelled on 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 G(R k ) of all linear subspaces of R k , such that where G 0 (R k ) = {0} and G k (R k ) = {R k } are trivial manifolds and G r (R k ) is a manifold modelled on the linear space R (k−r)×r for 0 < r < k. In consequence, G(R k ) is a manifold not modelled on a particular finite-dimensional space.
The atlas also endows M with a topology given by which makes (M, τ A ) a topological space where each local chart considered as a map between topological spaces, is a homeomorphism. 1

Main results and outline
Our first remark is that the matrix space R n×m is an analytic manifold modelled on itself and its geometric structure is fully compatible with the topology τ R n×m induced by a matrix norm. In this paper, we define an atlas on M r (R n×m ) which gives this set the structure of an analytic manifold, with a topology induced by the atlas fully compatible with the subspace topology τ R n×m | Mr(R n×m ) . This implies that M r (R n×m ) is an embedded submanifold of the matrix manifold R n×m modelled on itself 2 . For the topology τ R n×m , the matrix rank considered as a map is not continuous but only lower semi-continuous. However, if R n×m is seen as the disjoint union of sets of matrices with fixed rank, then R n×m has the structure of an analytic manifold not modelled on a particular finitedimensional space equipped with a topology which is not equivalent to τ R n×m , and for which the matrix rank is a continuous map.
Note that in the case when r = n = m, the set M n (R n×n ) coincides with the general linear group GL n of invertible matrices in R n×n , which is an analytic manifold trivially embedded in R n×n . In all other cases, which are addressed in this paper, our geometric description of M r (R n×m ) relies on a geometric description of the Grassmann manifold G r (R k ), with k = n or m.
Therefore, we start in Section 2 by introducing a geometric description of G r (R k ). A classical approach consists of describing G r (R k ) as the quotient manifold M r (R k×r )/GL r of equivalent classes of full-rank matrices Z in M r (R k×r ) having 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 = {(U Z , ϕ Z )} Z∈Mr(R k×r ) for G r (R k ), with local chart where Z ⊥ ∈ R k×(k−r) is such that Z T ⊥ Z = 0 and col k,r (A) denotes the column space of a matrix A ∈ R k×r , and we prove that the neighbourhood U Z have the structure of a Lie group. This parametrization of the Grassmann manifold is introduced in [2, 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 = {(V Z , ξ Z )} Z∈Mr(R k×r ) for the manifold M r (R k×r ) of matrices with full rank r < k, with local chart and prove that M r (R k×r ) is an analytic principal bundle with base G r (R k ) and typical fibre GL r . Moreover, we prove that M r (R k×r ) is an embedded submanifold of (R k×r , τ * R k×r ), and that each of the neighbourhoods V Z have the structure of a Lie group.
2 Note that the set M0(R n×m ) = {0} is a trivial manifold, which is trivially embedded in R n×m .
Finally, in Section 4, we provide an analytic atlas B n,m,r = {(U Z , θ Z )} Z∈Mr(R n×m ) for the set M r (R n×m ) of matrices Z = U GV T with rank r < min{n, m}, with local chart and we prove that M r (R n×m ) is an analytic principal bundle with base G r (R n )×G r (R m ) and typical fibre GL r . Then we prove that M r (R n×m ) is an embedded submanifold of (R n×m , τ * R n×m ), and that each of the neighbourhoods U Z have the structure of a Lie group.
2 The Grassmann manifold G r (R k ) In this section, we present a geometric description of the Grassmann manifold G r (R k ) of all subspaces of dimension r in R k , 0 < r < k, with an explicit description of local charts. We first introduce the surjective map where col k,r (Z) is the column space of the matrix Z, which is the subspace spanned by the column vectors of Z. Given V ∈ G r (R k ), there are infinitely many matrices Z such that col k,r (Z) = V. Given a matrix Z ∈ M r (R k×r ), the set of matrices in M r (R k×r ) having the same column space as Z is ZGL r := {ZG : G ∈ GL r }.

An atlas for G r (R k )
For a given matrix Z in M r (R k×r ), we let Z ⊥ ∈ M k−r (R k×(k−r) ) be a matrix such that Z T Z ⊥ = 0 and we introduce an affine cross section which has the following equivalent characterization.
Lemma 2.1. The affine cross section S Z is characterized by and the map Proof. We first observe that Z T (Z + Z ⊥ X Z ) = Z T Z for all X ∈ R (k−r)×r , which implies that {Z + Z ⊥ X : X ∈ R (k−r)×r } ⊂ S Z . For the other inclusion, we observe that if W ∈ S Z , then Z T W = Z T Z and hence W − Z ∈ col k,r (Z) ⊥ , the orthogonal subspace to holds, which means that the set of matrices with the same column space as W intersects Proof. Let us assume the existence of W,W ∈ S Z such that col k,r (W ) = col k,r (W ). Then W =W by Proposition 2.2.
Lemma 2.1 and Corollary 2.3 allow us to construct a system of local charts for G r (R k ) by defining for each Z ∈ M r (R k×r ) a neighbourhood of col k,r (Z) by together with the bijective map We denote by Z + the Moore-Penrose pseudo-inverse of the full rank matrix Z ∈ M r (R r×k ), defined by It satisfies Z + Z = id r and Z + Z ⊥ = 0. Moreover, ZZ + ∈ R k×k is the projection onto col k,r (Z) parallel to col k,r (Z) ⊥ . Finally, we have the following result.
is an open set. In the same way, we show that UZ = U Z ∩ UZ and ϕZ( , which is clearly an analytic map.

Lie group structure of neighbourhoods U Z
Here we prove that each neighbourhood U Z of G r (R k ) is a Lie group. For that, we first note that a neighbourhood U Z of G r (R k ) can be identified with the set S Z through the application col k,r : S Z → U Z . The next step is to identify S Z with a closed Lie subgroup of GL k , denoted by G Z , with associated Lie algebra g Z isomorphic to R r×(k−r) , and such that the exponential map 3 exp : g Z −→ G Z is a diffeomorphism. To this end, for a given Z ∈ M r (R k×r ), we introduce the vector space The following proposition proves that g Z is a commutative subalgebra of R k×k .
holds, and g Z is a commutative subalgebra of R k×k . Moreover, and Proof. Since (Z ⊥ XZ + )(Z ⊥X Z + ) = 0 holds for all X,X ∈ R (k−r)×r , the vector space g Z is a closed subalgebra of the matrix unitary algebra R k×k . As a consequence, (Z ⊥ XZ + ) p = 0 holds for all X ∈ R (k−r)×r and all p ≥ 2, which proves (6). We directly deduce (7) using ZZ + = id r , and (8) using Z + Z ⊥ = 0.
From Proposition 2.5 and the definition of S Z , we obtain the following results. 3 We recall that the matrix exponential exp : Corollary 2.6. The affine cross section S Z satisfies and for all X ∈ R (k−r)×r , where the brackets [·|·] are used for matrix concatenation.
Now we need to introduce the following definition and proposition (see [15, p.80]).
Definition 2.7. Let (K, +, ·) be a ring and let (K, +) be its additive group. A subset I ⊂ K is called a two-sided ideal (or simply an ideal) of K if it is an additive subgroup of K such that I·K := {r ·x : r ∈ I and x ∈ K} ⊂ I and K·I := {x·r : r ∈ I and x ∈ K} ⊂ I.
Proposition 2.8. If g ⊂ h is a two-sided ideal of the Lie algebra h of a group H, then the subgroup G ⊂ H generated by exp(g) = {exp(G) : G ∈ g} is normal and closed, with Lie algebra h.
From the above proposition, we deduce the following result.
Then g Z ⊂ R k×k is a two-sided ideal of the Lie algebra R k×k and hence is a closed Lie group with Lie algebra g Z . Furthermore, the map exp : which proves that R k×k ·g Z ⊂ g Z . This proves that g Z is a two-sided ideal. The map exp is clearly surjective. To prove that it is injective, we assume exp(Z ⊥ XZ + ) = exp(Z ⊥X Z + ) for X,X ∈ R (k−r)×r . Then from (6), we obtain Z + Z ⊥ X = Z + Z ⊥X and hence X =X, i.e. Z ⊥ XZ + = Z ⊥X Z + in g Z .
Finally, we can prove the following result.
Theorem 2.10. The set S Z together with the group operation × Z defined by for X,X ∈ R (k−r)×r is a Lie group.
Proof. To prove that it is a Lie group, we simply note that the multiplication and inversion maps It follows that U Z can be identified with a Lie group through the map ϕ Z .
is a Lie group and the map γ Z : U Z −→ G Z given by is a Lie group isomorphism.
3 The non-compact Stiefel principal bundle M r (R k×r ) In this section, we give a new geometric description of the set M r (R k×r ) of matrices with full rank r < k, which is based on the geometric description of the Grassmann manifold given in Section 2.

Principal bundle structure of M r (R k×r )
For Z ∈ M r (R k×r ), we define a neighbourhood of Z as From Proposition 2.2, we know that for a given matrix W ∈ V Z , there exists a unique pair of matrices (X, G) ∈ R (k−r)×r × GL r such that W = (Z + Z ⊥ X)G. Therefore, It allows us to introduce a parametrisation ξ −1 Z (see Figure 1) defined through the bijection such that Figure 1: Illustration of the chart ξ Z which associates to W = (Z + Z ⊥ X)G ∈ V Z ⊂ M r (R k×r ) the parameters (X, G) in R (k−r)×r × GL r .
Theorem 3.1. The collection B k,r := {(V Z , ξ Z ) : Z ∈ M r (R k×r )} is an analytic atlas for M r (R k×r ), and hence (M r (R k×r ), B k,r ) it is an analytic kr-dimensional manifold modelled on R (k−r)×r × R r×r .
Proof. {V Z } Z∈Mr(R k×r ) is clearly a covering of M r (R k×r ). Moreover, since ξ Z is bijective from V Z to R (k−r)×r × GL r we claim that if V Z ∩ VZ = ∅ for Z,Z ∈ M r (R k×r ), then the following statements hold: In this proof, we equip R k×r with the topology τ R k×r induced by matrix norms.
with ξ −1 Z (X, G) = (Z + Z ⊥ X)G, which is clearly an analytic map. 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 ∈ N ∪ {∞} or p = ω. In the latter case, C ω means analytic.
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 ψ • F • ϕ −1 is a representation of F using a system of local coordinates given by the charts (U, ϕ) and (W, ψ).
and let F be a manifold. A C p fibre bundle E with base B and typical fibre F is a C p manifold which is locally a product manifold, that is, there exists a surjective morphism π : E −→ B such that for each b ∈ B there is a C p diffeomorphism between manifolds In the case where F is a Lie group, we say that E is a C p principal bundle, and if F is a vector space, we say that it is a C p vector bundle.
Theorem 3.4. The set M r (R k×r ) is an analytic principal bundle with typical fibre GL r and base G r (R k ), with a surjective morphism between M r (R k×r ) and G r (R k ) given by the map col k,r .
Proof. To show that it is an analytic principal bundle, we first observe that is a surjective morphism. Indeed, let Z ∈ M r (R k×r ) and (V Z , ξ Z ) ∈ B k,r and (U Z , ϕ Z ) ∈ A k,r . Noting that col k,r (Y G) = col k,r (Y ) for all Y ∈ S Z , we obtain that col k,r (V Z ) = U Z .
Moreover, a representation of col k,r by using a system of local coordinates given by the charts is (ϕ Z • col k,r • ξ −1 Z )(X, G) = X, which is clearly an analytic map from R (k−r)×r × GL r to R (k−r)×r such that col −1 k,r (U Z ) = V Z . Now, a representation of the morphism using the system of local coordinates given by the charts is , which is clearly an analytic diffeomorphism. To conclude, consider the projection and observe that (p Z • χ Z )(W ) = col k,r (W ) holds for all W ∈ V Z .

M r (R k×r ) as a submanifold and its tangent space
Here, we prove that the non-compact Stiefel manifold M r (R k×r ) equipped with the topology given by the atlas B k,r is an embedded submanifold in R k×r . For that, we have to prove that the standard inclusion map as a morphism is an embedding. To see this we need to recall some definitions and results. 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 ∈ M, we denote by T m M the tangent space of M at m (in local coordinates).
is a map of class C p , where (U, ϕ) ∈ A is a chart in M containing m and (W, ψ) ∈ B is a chart in N containing F (m). Then we define For finite dimensional manifolds we have the following criterion for immersions (see Theorem 3.5.7 in [1]). A concept related to an immersion between manifolds is given in the following definition.  We first note that the representation of the inclusion map i using the system of local coordinates given by the charts Then the tangent map Proposition 3.10. The tangent map T Z i : R (k−r)×r × R r×r → R k×r at Z ∈ M r (R k×r ) is a linear isomorphism, with inverse (T Z i) −1 given by Proof. Let us assume that T Z i(Ẋ,Ġ) = Z ⊥Ẋ + ZĠ = 0. Multiplying this equality by Z + and Z + ⊥ on the left, we obtainĠ = 0 andẊ = 0 respectively, which implies that T Z i is injective. To prove that it is also surjective, we consider a matrixŻ ∈ R k×r and observe thatẊ = Z + ⊥Ż ∈ R (k−r)×r andĠ = Z +Ż ∈ R r×r is such that T Z i(Ẋ,Ġ) =Ż. Since T Z i is injective, the inclusion map i is an immersion. To prove that it is an embedding we equip M r (R k×r ) with the topology τ B k,r given by the atlas and we equip R k×r with the topology τ R k×r induced by matrix norms. We need to check that is a topological homeomorphism. Since the topology in (M r (R k×r ), τ B k,r ) has the property that each local chart ξ Z is indeed a homeomorphism from V Z in M r (R k×r ) to ξ Z (V Z ) = R (k−r)×r × GL r (see Section 1.1), we only need to show that the bijection Multiplying this equality by Z + on the left we obtainĠ = 0, and hence Z ⊥Ẋ G = 0. Multiplying by Z + ⊥ on the left we obtaiṅ XG = 0. ThusẊ = 0 and as a consequence D(i • ξ −1 Z )(X, G) is a linear isomorphism for each (X, G) ∈ R (k−r)×r × GL r . The inverse function theorem says us that (i • ξ −1 Z ) is a diffeomorphism, in particular a homeomorphism, and hence i is an embedding.
The tangent space to M r (R k×r ) at Z is the image through T Z i of the tangent space at Z in local coordinates T Z M r (R k×r ) = R (k−r)×r × R r×r , i.e.

Lie group structure of neighbourhoods V Z
We here prove that each neighbourhood V Z of M r (R k×r ) has the structure of a Lie group. For that, we first note that V Z can be identified with S Z × 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 × GL r , which is a Lie group with the group operation Z given by for X, X ∈ R (k−r)×r and G, G ∈ GL r . It allows us to define a group operation Z over and to state the following result.
Theorem 3.11. The set V Z together with the group operation Z defined by (15) is a Lie group and the map η Z : V Z −→ G Z × GL r given by 4 The principal bundle M r (R n×m ) for 0 < r < min(m, n) In this section, we give a geometric description of the set of matrices M r (R n×m ) with rank r < min(m, n).

M r (R n×m ) as a principal bundle
For Z ∈ M r (R n×m ), there exists U ∈ M r (R n×r ), V ∈ M r (R m×r ), and G ∈ GL r such that where the column space of Z is col n,r (U ) and the row space of Z is col m,r (V ). Let us first introduce the surjective map , col m,r (V )). The set −1 r (col n,r (U ), col m,r (V )) = {U HV T : H ∈ GL r } can be identified with GL r . Let us consider U ⊥ ∈ M n−r (R n×(n−r) ) such that U T U ⊥ = 0 and V ⊥ ∈ M m−r (R m×(m−r) ) such that V T V ⊥ = 0. Then we define a neighbourhood of U GV T in the set M r (R n×m ) by where U U and U V are the neighbourhoods of col n,r (U ) and col m,r (V ) respectively (see Section 2.2). Noting that U U = ϕ −1 U (R (n−r)×r ) = col n,r (S U ) and U V = ϕ −1 V (R (m−r)×r ) = 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 U GV T can be written We can associate to U Z the parametrisation θ −1 Z given by the chart (see Figure 2) Figure 2: Illustration of the chart θ Z which associates to Proof. {U Z } Z∈Mr(R n×m ) is clearly a covering of M r (R n×m ). Moreover, since θ Z is bijective from U Z to R (n−r)×r × R (m−r)×r × GL r , we claim that if U Z ∩ UZ = ∅ for Z = U GV T andZ =ŨGṼ T ∈ M r (R n×m ), then the following statements hold: In this proof, we equip R n×m with the topology τ R n×m induced by matrix norms. We first observe that the set U which is clearly an analytic map.
Theorem 4.2. The set M r (R n×m ) is an analytic principal bundle with typical fibre GL r and base G r (R n ) × G r (R m ) with surjective morphism r between M r (R n×m ) and G r (R n ) × G r (R m ) given by r .
Proof. To prove that it is an analytic principal bundle, we consider the surjective map the atlas A n,r := {(U U , ϕ U ) : U ∈ M r (R n×r )} of G r (R n ) and the atlas A m,r := Observe that for each fixed G ∈ GL r , we have that −1 is independent of the choice of Z = U GV T , where G ∈ GL r . Now, the representation of χ Z in local coordinates is the map , which is an analytic diffeomorphism. Moreover, let p Z : U U × U V × GL r −→ U U × U V be the projection over the two first components. Then (p Z • χ Z )(U HV T ) = (col n,r (U ), col m,r (V )) = r (U HV T ) and the theorem follows.

M r (R n×m ) as a submanifold and its tangent space
Here, we prove that M r (R n×m ) equipped with the topology given by the atlas B n,m,r is an embedded submanifold in R n×m . For that, we have to prove that the standard inclusion map i : M r (R n×m ) → R n×m is an embedding. Noting that the inclusion map restricted to the neighbourhood U Z of Z = U GV T is identified with Proposition 4.3. The tangent map T Z i : R (n−r)×r × R (m−r)×r × R r×r → R n×m at Z = U GV T ∈ M r (R n×m ) is a linear isomorphism with inverse (T Z i) −1 given by forŻ ∈ R n×m . Furthermore, the standard inclusion map i is an embedding from M r (R n×m ) to R n×m .
Proof. Let us suppose that T Z i(Ẋ,Ẏ ,Ḣ) = 0. Multiplying this equality by (U ⊥ ) + and U + on the left leads toẊ GV T = 0 and G(V ⊥Ẏ ) T +ḢV T = 0 respectively. By multiplying the first equation by (V + ) T on the right, we obtainẊ = 0. By multiplying the second equation on the right by (V + ) T and (V + ⊥ ) T , we respectively obtainḢ = 0 andẎ = 0. Then, T Z i is injective and then i is an immersion. Foṙ Z ∈ R n×m , we note thatẊ = U + Let us now equip M r (R n×m ) with the topology τ Bn,m,r given by the atlas and R n×m with the topology τ R n×m induced by matrix norms. We have to prove that i : (M r (R n×m ), τ Bn,m,r ) −→ (M r (R n×m ), τ R n×m |Mr(R n×m ) ) is a topological isomorphism. The topology in (M r (R n×m ), τ Bn,m,r ) is such that a local chart θ Z is a homeomorphism from U Z ⊂ M r (R n×m ) to θ Z (U Z ) = R (n−r)×r ×R (m−r)×r × GL r (see Section 1.1). Then, to prove that the map i is an embedding, we need to show that the bijection (i • θ −1 Z ) : R (n−r)×r × R (m−r)×r × GL r −→ U Z ⊂ R n×m is a topological homeomorphism. For that, observe that its differential D(i • θ −1 Z )(X, Y, H) ∈ L(R (n−r)×r × R (m−r)×r × R r×r , R n×m ) at (X, Y, H) ∈ R (n−r)×r × R (m−r)×r × GL r is given by Assume that Multiplying on the left by U + and on the right by (V + ) T , we obtainḢ = 0. Multiplying on the left by U + ⊥ and on the right by (V + ) T we deduce thatẊH = 0, that is,Ẋ = 0. Finally, multiplying on the left by U + and on the right by (V + ⊥ ) T we obtain HẎ T = 0, and henceẎ = 0. Thus, D(i • θ −1 Z )(X, Y, H) is a linear isomorphism from R (n−r)×r × R (m−r)×r × R r×r to D(i • θ −1 Z )(X, Y, H)[R (n−r)×r × R (m−r)×r × R r×r ] for each (X, Y, H) ∈ R (n−r)×r × R (m−r)×r × GL r . The inverse function theorem says us that (i • θ −1 Z ) is a diffeomorphism from R (n−r)×r ×R (m−r)×r ×GL r to U Z = (i•θ −1 Z )(R (n−r)×r ×R (m−r)×r × GL r ) and in particular, it is a topological homeomorphism. In consequence, the map i is an embedding.
The tangent space to M r (R n×m ) at Z = U GV T , which is the image through T Z i of the tangent space in local coordinates T Z M r (R n×m ) = R (n−r)×r × R (m−r)×r × R r×r , is T Z M r (R n×m ) = {U ⊥Ẋ GV T + U G(V ⊥Ẏ ) T + UĠV T :Ẋ ∈ R (n−r)×r ,Ẏ ∈ R (m−r)×r ,Ġ ∈ R r×r }, and can be decomposed into a vertical tangent space

Lie group structure of neighbourhoods U Z
We here prove that M r (R n×m ) has locally the structure of a Lie group by proving that the neighbourhoods U Z can be identified with Lie groups.
Let Z = U GV T ∈ M r (R n×m ). We first note that U Z can be identified with S U × S V × 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 U Z can be identified with a product of three Lie groups, which is a Lie group with the group operation Z given by (exp(U ⊥ XU + ), exp(V ⊥ Y V + ), G) Z (exp(U ⊥ X U + ), exp(V ⊥ Y V + ), G ) = (exp(U ⊥ (X + X )U + ), exp(V ⊥ (Y + Y )V + ), GG ).
It allows us to define a group operation Z over U Z defined for W = θ −1 Z (X, Y, G) and W = θ −1 Z (X , Y , G ) by and to state the following result. is a Lie group isomorphism.