Certain sets of matrices may be endowed a manifold structure, which can help us understand or solve problems. This article looks at some common matrix manifolds.
The generic, m-by-n matrix manifold $M_{m, n}(\mathbb{R})$ is the (m n)-dimensional real vector space of m-by-n real matrices, endowed with the inner product $\langle A, B \rangle = \text{tr}(A B^T)$. It is diffeomorphic to, and often identified with, the Euclidean (m n)-space: $M_{m, n}(\mathbb{R}) \cong \mathbb{R}^{mn}$. Probability models: matrix-variate Gaussian distribution $N_{n,k}(M; \Sigma_n, \Sigma_k)$. Full-rank manifold $M^∗_{m, n}(\mathbb{R})$ or $\mathbb{R}_∗^{mn}$ is the set of m-by-n full-rank real matrices, as an open subset of the m-by-n matrix manifold: $M^∗_{m, n}(\mathbb{R}) = \{X \in M_{m, n}(\mathbb{R}) : \text{rank}(X) = \min(m, n)\}$; or equivalently, $M^∗_{m, n}(\mathbb{R}) = M_{m, n}(\mathbb{R}) \setminus \bigcup_{k=0}^{\min(m,n)} \mathcal{M}(k, m \times n)$. It is a disconnected Riemannian (m n)-manifold. If m = n, its underlying set coincides with the general linear group of nonsingular order-n real matrices: $M^∗_n(\mathbb{R}) = \text{GL}(n, \mathbb{R})$. Alternatively, GLn can be defined as the product manifold of the orthogonal group and the positive-definite manifold: $\text{GL}_n = O(n) \times \mathcal{S}_+(n)$, because every nonsingular matrix has a unique polar decompostion, A = U P.
Rank-k manifold $\mathcal{M}(k, m \times n)$ is the set of m-by-n real matrices of rank $k \in \{0, \dots, \min(m, n)\}$, as an embedded Riemannian $k (m + n - k)$-submanifold of the m-by-n matrix manifold: $\mathcal{M}(k, m \times n) = \{X \in M_{m, n}(\mathbb{R}) : \text{rank}(X) = k\}$; its Riemannian metric is induced from the Euclidean metric of the m-by-n matrix manifold. Alternatively, it can be defined as a quotient manifold, with the order-k general linear group acting on the full-rank manifold of (m+n)-by-k matrices: $\mathcal{M}(k, m \times n) = M^∗_{m+n,k} / \text{GL}_k$, with equivalence class $[(M, N)] = {(M Q, N Q^{-T}) : Q \in \text{GL}_k}$, where $M \in M^∗_{m,k}$ and $N \in M^∗_{n,k}$. From this definition, it is easy to see that it has dimension $k (m + n - k)$. Riemannian metrics can be induced on it via the Riemannian submersion theorem: let $\pi: M^∗_{m,k} \times M^∗_{n,k} \mapsto \mathcal{M}(k, m \times n)$, $\pi(M, N) = M N^T$, which is a surjective smooth submersion, consider the fiber bundle $(M^∗_{m+n,k}, \pi)$ with the (aforementioned) action of group $\text{GL}_k$ and an arbitrary Riemannian metric, there is a unique Riemannian metric on $\mathcal{M}(k, m \times n)$ such that π is a Riemannian submersion (see e.g. [@Absil2014]).
Symmetric matrix manifold $\mathcal{S}(n)$ is the set of order-n real symmetric matrices, as the n(n+1)/2-dimensional subspace of the order-n matrix manifold: $\mathcal{S}(n) = \{X \in M_n(\mathbb{R}) : X = X^T\}$. Probability models: symmetric Gaussian distribution $N_{n,n}(M; \Sigma)$. Positive-semidefinite manifold $\mathcal{S}_{\ge 0}(n)$ is the order-n positive semi-definite cone, as a regular domain of the symmetric matrix manifold: $\mathcal{S}_{\ge 0}(n) = \{X \in \mathcal{S}(n) : X \ge 0\}$.
Positive-definite manifold $\mathcal{S}_+(n)$, aka positive symmetric cone, is the set of order-n positive-definite matrices, as an open convex subset of the symmetric matrix manifold (with the Euclidean metric): $\mathcal{S}_+(n) = \{X \in \mathcal{S}(n) : X > 0\}$. Alternatively, it can be defined as a quotient manifold, with the orthogonal group acting on the general linear group: $\mathcal{S}_+(n) = \text{GL}_n / O(n)$, with identification $C = A A^T$, where $C \in \mathcal{S}_+(n)$ and $A \in \text{GL}_n$, and equivalence class $[A] = {(A Q : Q \in O(n)}$. Since O(n) is a subgroup of GL_n, the set GL_n / O(n) with the action of GL_n is a homogeneous space. In fact, it is a reductive homogeneous space (see e.g. [@Bonnabel2010, A.1]), and there is a unique Riemannian metric that is invariant under the action of GL_n (by congruence): $g_C(T_1, T_2) = \text{tr}(T_1 C^{-1} T_2 C^{-1})$; with action $A \cdot C = A C A^T$ and $A = C^{-1/2}$, one can transport any C to the identity I, and any tangent vector at C to a tangent vector at I; this metric coincide with the Euclidean metric at the identity, $g_I(T_1, T_2) = \text{tr}(T_1 T_2)$. This metric is known as the natural metric [@Faraut1994; @Bhatia2007]; up to a scaling factor, it is also known as the Siegel metric in symplectic geometry, or the affine-invariant metric; in statistics, it is the Fisher information metric for the multivariate Gaussian [@Smith2005]; in optimization, it is the metric defined by the natural self-concordant logarithmically homogeneous barrier [@Boyd2004]. The quotient manifold with the natural metric has the following properties: geodesically complete, the exponential map is $\exp_C(T) = C^{1/2} \exp(C^{-1/2} T C^{-1/2}) C^{1/2}$, in particular, the exponential map at I is the matrix exponential, $\exp_I(T) = \exp(T)$; the geodesic distance is $d_g(C, D) = \|\log(C^{-1/2} D C^{-1/2})\|_F$, computed as |log λ| where λ are the generalized eigenvalues of C - λ D, and is invariant under matrix inversion $d_g(C, D) = d_g(C^{-1}, D^{-1})$; a Riemannian symmetric space, where every point C has a point reflection defined by $\sigma^C(D) = C D^{-1} C$ [@Bhatia2007, p. 226]; sectional curvature is nonpositive, and therefore every compact set has a unique center of mass; the intrinsic mean of two matrices is the midpoint of the geodesic connecting A and B, aka the geometric mean, $A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2}$. [@Arsigny2007] introduced the log-Euclidean metric, and the induced geometric mean is the matrix exponential of the arithmetic mean of matrix logarithms, which is easier to compute than the geometric mean under the natural metric. The positive-definite manifold can also be identified with the manifold $\mathcal{L}_{+}(n)$ of (lower) triangular matrices with positive diagonal entries, due to the uniquenss of Cholesky decomposition; and [@LinZH2019] introduced the log-Cholesky metric induced from a bi-invariant metric on $\mathcal{L}_+(n)$, which is even easier to compute and numerically more stable, and the parallel transport has a closed form and is easy to compute. Probability models: (noncentral) Wishart distribution $W_n(m, \Sigma; \Omega)$.
Rank-k positive-semidefinite manifold $\mathcal{S}_{\ge 0}(k, n)$ is the set of rank-k order-n positive-semidefinite matrices, as a Riemannian $k (2n - k + 1) / 2$-submanifold of the order-n matrix manifold: $\mathcal{S}_{\ge 0}(k, n) = \mathcal{S}_{\ge 0}(n) \cap \mathcal{M}(k, n \times n)$. It is related to the product manifold of the Stiefel manifold and the non-increasing positive Euclidean k-space by a surjective map: $f: V_{k, n} \times \mathbb{R}^k_{+\downarrow} \mapsto \mathcal{S}_{\ge 0}(k, n)$, $f(V, \lambda) = V \text{diag}(\lambda) V^T$, where $\mathbb{R}^k_{+\downarrow} = \{x \in \mathbb{R}^k_+ : \forall i < j, x_i \ge x_j\}$. Notice that the domain and codomain have the same manifold dimension. This also applies to $\mathcal{S}_+(n)$ with k = n. The natural metric on $\mathcal{S}_+(n)$ can be extended to $\mathcal{S}_{\ge 0}(k, n)$ as the sum of the natural metric on $\mathcal{S}_+(k)$ and the standard metric of $G_{k,n}$, which is invariant with respect to transformations that preserve angles (e.g. orthogonal transformations, scalings, and pseudoinversion), and the resulting manifold is geodesically complete [@Bonnabel2010]. The associated Riemannian distance can be approximately computed via SVD. The induced geometric mean preserves the rank, and is easy to compute.
Rank-k symmetric projection manifold $\mathcal{P}(k, n)$ is the set of rank-k symmetric projection matrices, as an embedded Riemannian k(n-k)-submanifold: $\mathcal{P}(k, n) = \{P \in \mathcal{S}(n) : P^2 = P, \text{rank}(P) = k\}$. Its smooth manifold structure is derived from Lie group theory: because O(n) acts smoothly on $\mathcal{S}(n)$ by the map $\phi: O(n) \times \mathcal{S}(n) \mapsto \mathcal{S}(n)$ defined by $\phi(Q, S) = Q S Q^T$, the orbit $O(n) \cdot P_k = \mathcal{P}(k,n)$ of the point $P_k = I_{n,k} I_{k,n} = (I_k, 0; 0, 0)$ is a properly embedded submanifold of $\mathcal{S}(n)$, and it is diffeomorphic to the quotient manifold $O(n) / (O(n - k) \times O(k))$; here $O(n - k) \times O(k)$ is used in place of $\{\text{diag}(R_1, R_2) : R_1 \in O(k), R_2 \in O(n-k)\}$, the isotropy group of $P_k$, because they are diffeomorphic. Its Riemannian metric is induced from the Euclidean metric of the order-n matrix manifold. With this metric, the surjective smooth submersion $\pi: O(n) \mapsto \mathcal{P}(k,n)$ defined by $\pi(Q) = \phi(Q, P_k)$ is a Riemannian submersion. Its tangent space can be written as: $T_P \mathcal{P(k,n)} = \{Q (0, K^T; K, 0) Q^T : K \in M_{n-k,k}\}$, where Q is an orthogonal representation of P such that $\pi(Q) = P$; note that the tangent vectors are symmetric matrices (i.e. in the tangent space of $\mathcal{S}(n)$). The horizontal tangent space at Q can be written as: $H^\pi_Q O(n) = \{Q (0, -K^T; K, 0) : K \in M_{n-k,k}\}$. The vertical tangent space at Q can be written as: $V^\pi_Q O(n) = \{Q (\Omega, 0; 0, \bar{\Omega}) : \Omega \in \Omega(k), \bar{\Omega} \in \Omega(n-k)\}$. Note that the right-hand side matrices are anti-symmetric (i.e. the outcome is in the tangent space of O(n)). The horizontal lift of tangent vector $Z \in T_P \mathcal{P}(k,n)$ to Q can be calculated as: $Z_Q = [Z, P] Q = (Z P - P Z) Q$. With the surjective smooth submersion $\pi: V_{k,n} \mapsto \mathcal{P}(k,n)$ defined by $\pi(X) = X X^T$, the horizontal lift of tangent vector Z to X can be calculated as $Z_X = Z X$. It is a Riemannian symmetric space, and every point P has a point reflection $\sigma^P: \mathcal{P}(k,n) \mapsto \mathcal{P}(k,n)$ defined by $\sigma^P(\tilde{P}) = S \tilde{P} S^T$, where $S = Q S_k Q^T$, $P = Q P_k Q^T$ is an eigen-decomposition, and $S_k = \text{diag}(I_k, -I_{n-k})$. Its Riemann (1,3)-curvature tensor at $P_k$ is $R(x, y) z = b$, where $x, y, z, b \in T_{P_k} \mathcal{P}(n,k)$ have the form $x = (0, X^T; X, 0)$ with $X \in M_{n-k,k}$, and $B = Z X^T Y - Z Y^T X - X Y^T Z + Y X^T Z$. When $k (n-k) \ge 2$, its sectional curvature at a point P is: $K_P(Z_1, Z_2) = 4 (\text{tr}(Z_1^2 Z_2^2) - \text{tr}((Z_1 Z_2)^2)) / (\text{tr}(Z_1^2) \text{tr}(Z_2^2) - \text{tr}^2(Z_1 Z_2)))$; or more compactly, using the Euclidean metric and the bracket, $K_P(Z_1, Z_2) = 2 \| [Z_1, Z_2] \|_F^2 / (\|Z_1\|_F^2 \|Z_2\|_F^2 - \langle Z_1, Z_2 \rangle_0^2)$. For n = 2 and k = 1, the sectional curvature is undefined. For n > 2 and k = 1 or n-1, the sectional curvature is always 1. For all other cases, the sectional curvature has a range [0, 2], see e.g. [@WongYC1968]. Integral of the invariant measure on the manifold [@Chikuse2003, Eq 1.4.11]: $g(k, n) := \int_{\mathcal{P}(k,n)} (d P) = \pi^{k(n-k)/2} \Gamma_k(k/2) \Gamma_k^{-1}(n/2)$.
Skew symmetric matrix manifold $\Omega(n)$ is the set of order-n real skew symmetric matrices, as the embedded Riemannian n(n-1)/2-submanifold of the order-n matrix manifold: $\Omega(n) = \{X \in M_n(\mathbb{R}) : X = - X^T\}$.
Other matrix manifolds:
Stiefel manifold $V_{k, n}$ is the collection of orthonormal k-frames in the Euclidean n-space, as an embedded Riemannian submanifold of the m-by-k matrix manifold: $V_{k, n} = \{X \in M_{n,k}(\mathbb{R}) : X^T X = I_k\}$, $k \in \{1, \dots, n\}$. It is a submersion level set $F(X) = X^T X - I_k = 0$, and therefore an embedded submanifold [@Absil2008, 3.3.2]. Riemannian metric is induced from Euclidean metric: $g_X(Z, W) = \text{tr}(Z^T W)$. The smooth manifold structure can also be derived from Lie group theory: because O(n) acts smoothly on $M_{n,k}$, the orbit $O(n) \cdot I_{n,p} = V_{k,n}$ of the point $I_{n,p} = (I_p; 0)$ is a properly embedded submanifold of $M_{n,k}$, and it is diffeomorphic to the quotient manifold $O(n) / O(n - k)$; here O(n - k) is used in place of the isotropy group $\{\text{diag}(I_k, R) : R \in O(n-k)\}$ because they are diffeomorphic. If k = 1, it is the (n-1)-sphere: $V_{1,n} = \mathbb{S}^{n-1}$, with dimension (n - 1). If k = n, it coincides with the orthogonal group: $V_{n,n} = O(n)$, with dimension n (n - 1) / 2.
Manifold property. Dimension, k (2n - k - 1) / 2. For O(n), n (n-1) / 2; for (n-1)-sphere, n-1. Compact, because it is diffeomorphic to the quotient manifold $O(n) / O(n-k)$. Connectedness: (n-k-1)-connected (see n-connected space). When k = n, it is O(n), which has two components, with determinant 1 for SO(n) and -1 for the other component. When k = n-1, it is path-connected. When k < n-1, it is simply connected. Homogeneous, because O(n) acts transitively on the left. Complete (if k = n, both components are metrically complete). Isometry group includes O(n) acting on the left, and O(k) acting on the right. Orientable. Integral of the invariant measure on the Stiefel manifold [@Chikuse2003, Eq 1.4.8]: $v(k, n) := \int_{V_{k,n}} (d X) = 2^k \pi^{kn/2} \Gamma_k^{-1}(n/2)$.
Constructs as a submanifold. Tangent space has the following equivalent forms: (1) as linear constraints, $T_X V_{k, n} = \{Z \in M_{n,k} : X^T Z + Z^T X = 0\}$; (2) in a "constructive" form, $T_X V_{k, n} = \{X \Omega + X_\perp K : \Omega \in \Omega(k), K \in M_{n-k,k} \}$, given an arbitrary orthogonal completion $(X, X_\perp) \in O(n)$; (3) in a projective form (aka tangential projection), $T_X V_{k, n} = \{X~\text{skew}(X^T M) + (I_n - X X^T) M : M \in M_{n,k}\}$, where $\text{skew}(A) = (A - A^T) / 2$. For orthogonal group, $T_Q O(n) = \{Q \Omega : \Omega \in \Omega(n)\}$. Normal space (w.r.t. the canonical metric): (1) in a constructive form, $N_X V_{k, n} = \{X S : S \in \mathcal{S}(k)\}$; (3) in a projective form, $N_X V_{k, n} = \{X~\text{sym}(X^T M) : M \in M_{n,k}\}$, where $\text{sym}(A) = (A + A^T) / 2$.
Canonical metric. The Riemannian metric of the orthogonal group is scaled by 1/2 in some conventions [@Zimmermann2017; @Bendokat2020]: $g_Q(Z, W) = 1/2~\text{tr}(Z^T W)$, where $Z, W \in T_Q O(n)$; because tangent vectors of O(n) has the form $Q \Omega$ with $\Omega \in \Omega(n)$, the metric can also be written as $g_Q(Q \Omega_1, Q \Omega_2) = 1/2~\text{tr}(\Omega_1^T \Omega_2)$. This does not change the geometry of O(n): the tangent vectors are scaled by $1/\sqrt{2}$, and so is the Riemannian distance function. The "canonical metric" on the Stiefel manifold is induced from the orthogonal group via a smooth submersion $\pi_k: O(n) \mapsto V_{k,n}$, which takes an order-n matrix to its first k columns: $\pi_k(Q) = Q I_{n,k} = (q_i)_{i=1}^k$. The horizontal tangent space at Q can be written as: $H^{\pi_k}_Q O(n) = \{Q (\Omega, -K^T; K, 0) : \Omega \in O(k), K \in M_{n-k,k}\}$. The vertical tangent space at Q can be written as: $V^{\pi_k}_Q O(n) = \{Q (0, 0; 0, \bar{\Omega}) :\bar{\Omega} \in O(n-k)\}$. Note that the right-hand side matrices are anti-symmetric (i.e. the outcome is in the tangent space of O(n)). For a tangent vector $Z = X \Omega + X_\perp K \in T_X V_{k,n}$, its horizontal lift to $Q \in O(n)$ via $\pi_k$ can be written as $Z_Q = Q \tilde{\Omega}$, where $\tilde{\Omega} = (\Omega, -K^T; K, 0)$. Therefore the canonical metric of the Stiefel manifold is $g_X(Z_1, Z_2) = g_Q(Z_{1,Q}, Z_{2,Q})$, which equals $1/2~\text{tr}(\Omega_1^T \Omega_2) + \text{tr}(K_1^T K_2)$, and can be computed as $g_X(Z_1, Z_2) = \text{tr}(Z_1^T (I_n - 1/2 X X^T) Z_2)$. Note that the metric on O(n) is consistent with the canonical metric of the Stiefel manifold. The canonical metric gives the Stiefel manifold a different geometry than that of a Riemannian submanifold of a Euclidean space.
Geometric operations. Riemannian metric: $g_X(Z, W) = \text{tr}(Z^T W) = \text{tr}(\Omega_Z^T \Omega_W + K_Z^T K_W)$, where $Z = X \Omega_Z + X_\perp K_Z$ and $W = X \Omega_W + X_\perp K_W$. Covariant derivative (w.r.t. the Levi-Civita / tangential connection): $\nabla_Z W(X) = P_X (\bar{\nabla}_Z W(X))$, where $Z \in T_X V_{k, n}$, $W \in \mathfrak{X}(V_{k, n})$. Riemannian distance function: closed form unknown? Exponential map: $\exp_X(Z) = (X, Z) \exp([X^T Z, -Z^T Z; I_k, X^T Z]) (I_k; 0) \exp(-X^T Z)$ (uses matrix exponential). Parallel transport: closed form unknown. Retractions (mostly based on matrix decompositions): (1) QR decomposition: $R(X, Z) = Q$, where $X + Z = Q R$, R has positive diagonal elements (modified Gram-Schmidt, about $2nk^2$ FLOPs). (2) Polar decomposition: $R(X, Z) = (X + Z) (I_k + Z^T Z)^{-1/2}$ (iterative $O(k^3)$ for eigen-decomposition + $O(nk^2)$ scalar additions and multiplications). Vector transports (associated with a retraction): (1) by differentiated retraction: $T_W(Z) = R \rho(R^T Z R') + (I - R R^T) Z R'$, where $R = R(X, W)$ is the QR-based retraction, $R' = (R^T (X + W))^{-1}$, and $\rho(A) = L - L^T$ (L is the lower triangle of A); (2) by tangential projection (as a submanifold): $T_W(Z) = (I - Y Y^T) Z + Y~\text{skew}(Y^T Z)$, where $Y = R(X, W)$ is any retraction;
Sectional curvature has a range [0, 5/4].
Probability models [@Chikuse2003]: matrix Langevin $L_{k,n}(F)$, aka matrix von Mises-Fisher (vMF), exponential-linear; matrix Bingham $B_{k,n}(B)$, exponential-quadratic; matrix angular central Gaussian $\text{MACG}(\Sigma)$; max entropy with moment constraints [@Pennec2006]. Sampling: geodesic Monte Carlo is applicable [@Byrne2013].
Grassmann manifold or Grassmannian $G_{k, n}$ or $G_k(\mathbb{R}^n)$ is an (abstract) Riemannian manifold, defined as follows: its underlying set consists of k-subspaces of the Euclidean n-space, $G_{k, n} = \{\text{Span}(M) : M \in M^∗_{n, k}\}$, $k \in \{1, \dots, n\}$; its smooth manifold structure is uniquely determined such that group $\text{GL}_n$ is a smooth left action on $G_{k,n}$ [@Lee2012, Ex 21.21]; its Riemannian metric is uniquely determined such that $\text{Span}: V_{k,n} \mapsto G_{k,n}$ is a Riemannian submersion, because the orthogonal group O(k) is an isometric right action on the Stiefel manifold $V_{k,n}$ [@Lee2018, Prob 2-7]. If k = 1, it is the (n-1)-dimensional projective space, i.e. the quotient manifold of lines in the Euclidean n-space: $G_{1,n} = \mathbb{RP}^{n-1}$. If k = n, it is a singleton consisting of the Euclidean n-space: $G_{n,n} = \{\mathbb{R}^n\}$. Note that $G_{k,n}$ and $G_{n-k,n}$ can be identified, because orthgonal complement $\perp: G_{k,n} \mapsto G_{n-k,n}$ is an isometry. Although the Grassmann manifold consists of subspaces rather than matrices, it is closely related to matrix manifolds and sometimes identified with $\mathcal{P}(k,n)$.
Manifold property. Dimension, k (n - k). Compact, because it is diffeomorphic to the quotient manifolds $V_{k, n} / O(k)$ [@Lee2012, Prob 21-13] and $O(n) / (O(n-k) \times O(k))$. Simply connected, except $G_{1, 2} \cong \mathbb{S}^1$; (see e.g. this paper) Symmetric, isotropic (and therefore homogeneous) [@Lee2018, Prob 3-20]. Complete. Isometry group includes O(n) on the left, and $N_G(H)/H$ on the right, where $N_G(H)$ denotes the normalizer of H in G, with G = O(n) and isotropy group $H = O(n-k) \times O(k)$. Orientable if and only if n is even. (Note that "oriented Grassmann manifold" means the set of oriented subspaces, which does not mean the Grassmann manifold itself is oriented.)
Representations of the abstract manifold. Because $\text{Span}: V_{k,n} \mapsto G_{k,n}$ and $\text{Span}: M^∗_{n, k} \mapsto G_{k,n}$ are Riemannian submersions, one may use $M \in M^∗_{n, k}$ as a representation of Span(M) to carry out all the computations related to the Grassmann manifold. Its equivalence class is $[M] = \{M A : A \in \text{GL}_k\}$; for $X \in V_{k, n}$, $[X] = \{X Q : Q \in O(k)\}$. For notational simplicity, we may use the equivalence class [M] and abstract element Span(M) interchangeably, when there is no ambiguity. Horizontal tangent space (represents the abstract tangent space; normal space to the fiber at the representation) has the following equivalent forms: (1) as linear constraints, $H_M = \{Z \in M_{n, k} : M^T Z = 0\}$; (2) in a "constructive" form, $H_X = \{X_\perp B : B \in M_{n-k,k}\}$ given an arbitrary orthogonal completion $(X, X_\perp) \in O(n)$; (3) in a projective form (aka horizontal projection), $H_M = \{(I_n - M (M^T M)^{-1} M^T) Z : Z \in M_{n,k}\}$, and for Stiefel representations, $H_X = \{(I_n - X X^T) Z : Z \in M_{n,k}\}$. With two representations M and M' of the same abstract element [M] such that M' = M A where $A \in \text{GL}_k$, if $Z_M, Z_{M'}$ represent the same abstract tangent vector $Z \in T_{[M]} G_{k, n}$, then $Z_{M'} = Z_M A$. Vertical tangent space (analogous to normal space of a submanifold, tangent space to the fiber at the representation) $V_M = \{M A : A \in M_{k, k}\}$; for Stiefel representations, $V_X = \{X \Omega : \Omega \in \Omega(k)\}$. Note that by definition, $V_X = T_X [X]$ and $H_X = V_X^\perp$ w.r.t. $T_X V_{k,n}$.
Representations of an explicit identification. One may identify the underlying sets of the Grassmannian and the rank-k symmetric projection manifold, because $\text{Range}: \mathcal{P}(k, n) \mapsto G_{k,n}$ is a bijection, and so is the map $P: G_{k,n} \mapsto \mathcal{P}(k, n)$ that takes a subspace to its orthogonal projection operator. Let quotient map $\pi: V_{k,n} \mapsto \mathcal{P}(k,n)$ with $\pi(X) = X X^T$. Let quotient map $\pi_k: O(n) \mapsto V_{k,n}$ with $\pi_k(Q) = Q I_{n,k} = (x_i)_{i=1}^k$, which takes the first k columns of a matrix. Without ambiguity, let $\pi: O(n) \mapsto \mathcal{P}(k,n)$ with $\pi(Q) = \pi(\pi_k(Q))$; note that $\pi(Q) = \phi(Q, P_k)$. These quotient maps π are surjective smooth submersions, and as Riemannian submersions, the induced Riemannian metric coincides with (1/2 of) the (restricted) Euclidean metric. One may use $X \in V_{k,n}$ or $Q \in O(n)$ as a representation of $\pi(X) = \pi(Q) \in \mathcal{P}(k,n)$, see e.g. [@Bendokat2020, Fig 2.1].
Geometric operations. In the following, an abstract element of the Grassmannnian (e.g. p) is replaced by a basis or Stiefel representation (e.g. M, X) of the subspace (e.g. [M], [X]), and an abstract tangent vector (e.g. v, Δ) is replaced by its horizontal lift to the corresponding representation (e.g. Z, W). Riemannian metric $g_M(Z, W) = \text{tr}((M^T M)^{-1} Z^T W)$, where $Z, W \in H_M$; for a Stiefel representation, $g_X(Z, W) = \text{tr}(Z^T W)$, which is the same as the Euclidean metric [@Absil2008, Prop 3.4.6, Sec 3.6.2]. Covariant derivative (w.r.t. the Levi-Civita connection) $\overline{\nabla_Z W(X)} = P_X (\bar{\nabla}_Z W(X))$. Exponential map: $\exp_M(Z) = \pi_p(M) W \cos(\Sigma) + U \sin(\Sigma)$; where $Z = U \Sigma W^T$ is a thin SVD, and $\pi_p(M) = M (M^T M)^{-1/2}$ is the projection based on polar decomposition, which can be computed as $\pi_p(M) = A B$ via thin SVD $M = A S B$; for a Stiefel representation, $\exp_X(Z) = X W \cos(\Sigma) + U \sin(\Sigma)$. Parallel transport along geodesics: given $X \in V_{k, n}$ and $Z \in H_X$, denote $Y = \exp_X(Z)$, the parallel transport map $P^Z_{XY}: H_X \cong H_Y$ can be computed as $P^Z_{XY} = (-X W \sin(\Sigma) + U \cos(\Sigma)) U^T + (I - U U^T)$, where $Z = U \Sigma W^T$ is a thin SVD. Retractions: $R_X(Z) = X + Z$. Vector transports (associated with a retraction): $T^W_{XY} = P_Y$ where $Y = X + W$; the same result by differentiated retraction and by horizontal projection. Principal angles between two subspaces: (1) given Stiefel representations X and Y, $\theta = \arccos(\sigma(X^T Y))$; (2) given the horizontal lift $Z = \log_X(Y) \in H_X$ of the Grassmann logarithm, $\theta = \text{rev}(\sigma(Z))$, where the singular values are reversed because $\theta$ is in non-decreasing order by convention. Riemannian distance function: $d_g([X], [Y]) = \|\theta([X], [Y])\|$. Note that the projection metric (or gap metric) $\|X^T X - Y^T Y\|_2 = \sin \theta_k$. Curvature: see $\mathcal{P}(k, n)$.
Cut locus and conjugate locus. The tangent cut locus and the tangent conjugate locus of any point p on a Grassmann manifold have simple characterizations on the horizontal tangent space of a Stiefel representation X, both expressed in terms of their singular values. The horizontal lift of the tangent cut locus of p is a sphere of radius π/2 in spectral norm: $\text{TCL}(p)_X = \{Z \in H_X : \sigma_1(Z) = \pi / 2\}$. The horizontal lift of the tangent conjugate locus of p corresponds to a countable collection of planes in the space of singular values: (1) if k ≠ n/2, then $\text{Conj}(p)_X = \{Z \in H_X : \sigma(Z) \in S_1 \cup S_2 \}$, where $S_1 = \cup_{i=1}^u \cup_{m=1}^\infty \{\sigma \in S : \sigma_i = m \pi\}$, $S_2 = \cup_{1 \le i < j \le u} \cup_{m=1}^\infty \{\sigma \in S : \sigma_i - \sigma_j = m \pi\}$, and $S = \mathbb{R}^u_{+\downarrow}$, where u = k if k ≤ n/2, and u = n-k if k > n/2; (2) if k = n/2, $S_1$ should be removed. Note that this is essentially proved in [@Bendokat2020, Thm 7.2], but they use conjugate points without reference to geodesics or tangent vectors, which is very confusing. For Grassmann manifolds, an abstract tangent vector has the same length as its horizotnal lift (w.r.t. the Euclidean metric not the canonical metric): let $\Delta \in T_p G_{k,n}$, then $\|\Delta\| = \|\Delta_X\|_F = \|\sigma(\Delta_X)\|$. Therefore, length is preserved if we map the horizontal tangent space $H_X$ to the space $S = \mathbb{R}^u_{+\downarrow}$ of singular values. Within this space, the (tangent) cut locus corresponds exactly to the (u-1)-plane, $\sigma_1 = \pi/2$; and the tangent conjugate locus corresponds exactly to the collection of planes, $S_1 \cup S_2$. In particular, the tangent cut locus of a Grassmann manifold always comes before the tangent conjugate locus, if the latter exists. Therefore, the Grassmann manifold at any point has injectivity radius: inj(p) = π/2; and the maximum distance between any two points is: $\max d_g(p, p') = \sqrt{u} \pi / 2$. The cardinality of shortest geodesic to a cut point is |O(r)|, where r denotes the number of principal angles equal to π/2.
Riemannian logarithm. Given Stiefel representations X and Y for $p, p' \in G_{k,n}$, let $\Delta_X \in H_X$ be the horizontal lift of the generalized Grassmann logarithm $\Delta = \log_p(p')$ to X. A modified algorithm for the (generalized) Grassmann logarithm computes $\Delta_X$, first described in [@Zimmermann2019] with properties in [@Bendokat2020, Thm 5.4, 5.5]: $Y^T X = \tilde{Q} \tilde{S} \tilde{R}^T$, an order-k SVD; $\bar{Y} = Y (\tilde{Q} \tilde{R}^T)$ and $\bar{X} = X (\tilde{R} \tilde{S} \tilde{R}^T)$; $\bar{Y} - \bar{X} = \hat{Q} \hat{S} \hat{R}^T$, an n-by-k thin SVD. Output: let $r = \sum_{i=1}^k 1(\hat{s}_i = 1)$, if r = 0, then $p' \notin \text{Cut}(p)$, and $\Delta_X = \hat{Q} \arcsin(\hat{S}) \hat{R}^T$, which retains the Stiefel representative: $Y = \exp_X(\Delta_X)$; if r > 0, then $p' \in \text{Cut}(p)$, and $\Delta_X = \{\hat{Q} \arcsin(\hat{S}) \text{diag}(W, I_{k−r}) \hat{R}^T : W \in O(r)\}$. Notice that many other quantities can be computed with intermeidate results in this algorithm: (1) principal angles $\theta = \arccos \tilde{s} = \text{rev}(\arcsin \hat{s})$, where rev() denotes reversing the order of a vector; in particular, the singular values of the horizontal lift $\Delta_X$ of the (generalized) Grassmann logarithm are the principal angles (in non-increasing order), $\theta = \text{rev}(\sigma(\Delta_X))$; (2) principal bases $\tilde{X} = X \tilde{Q}$ and $\tilde{Y} = Y \tilde{R}$; (3) Riemannian distance $d_g(p, p') = \|\theta\|$.
Probability models [@Chikuse2003]: matrix Langevin $L^{(P)}_{k,n}(B)$, uses trace; matrix angular central Gaussian $\text{MACG}(\Sigma)$, uses (order-k) determinant.