Mathematics

Questions tagged [stiefel-manifolds]

For questions about Stiefel-manifolds, the set of all orthonormal $k$-frames in $\Bbb R^n$.

78 questions
19
votes
1 answer

Fundamental groups of Grassmann and Stiefel manifolds

Could someone provide details on how to compute fundamental groups of real and complex Grassmann and Stiefel manifolds?
google
  • 193
14
votes
1 answer

Difference between Grassmann and Stiefel manifolds

I'm working on an optimization problem on manifolds and I'm having a bit of a conceptual issue with choosing between the Grassmann and Stiefel manifolds. Grassmann(2, 3) is the linear subspace of dimension 2 within the space $\mathbb{R}^3$, so all…
bibliolytic
  • 222
9
votes
1 answer

Is the Stiefel manifold $V_k(\mathbb{R}^n)$ homeomorphic to $O(n)/O(n-k)$?

Am I right that the Stiefel manifold $V_k(\mathbb{R}^n)$ (set of all orthonormal $k$ -frames in $\mathbb{R}^n$ ) is homeomorphic to $O(n)/O(n-k)$ ? Sketch of proof: any element of $V_k(\mathbb{R}^n)$ can be obtained from standard basis…
Aspirin
  • 5,919
9
votes
1 answer

Obstruction cocycle of Stiefel manifold

I was reading about the interpretation of Stiefel Whitney classes as obstructions from Milnor-Stasheff's book and I got stuck at a step. The context is the following. Let $E \to B$ be a vector bundle of rank $n$ and let $V_k(\mathbb{R}^n)$ be the…
Manuel
  • 1,788
7
votes
2 answers

Nearest semi-orthogonal matrix using the entry-wise $ {\ell}_{1} $ norm

Given an $m \times n$ matrix $M$ ( $m \geq n$ ), the nearest semi-orthogonal matrix problem in $m \times n$ matrix $R$ is $$\begin{array}{ll} \text{minimize} & \| M - R \|_F\\ \text{subject to} & R^T R = I_n\end{array}$$ A solution can be found by…
Francis
  • 823
7
votes
1 answer

Geodesic on Stiefel manifold

Define a metric on Stiefel manifold $V_{n,p}$ as $$\langle \Delta_1,\Delta_2 \rangle = \operatorname{tr} \left( \Delta_1^T\left(I-\frac{1}{2}YY^T\right)\Delta_2 \right)$$ for all $\Delta_1, \Delta_2 \in T_Y V_{n,p}$ . How to calculate the geodesic…
gaoxinge
  • 4,652
6
votes
1 answer

Orientability of Stiefel manifold $V_2(\mathbb R^4)$

What is an easy proof of orientability of Stiefel manifold $V_2(\mathbb{R}^4)$ (pairs of orthonormal vectors from $\mathbb{R}^4$ - subset of $\mathbb{R}^8$)? All proofs I found deal with Lie groups and other complicated for me stuff. I suppose that…
Richard Wagner
  • 136
5
votes
1 answer

Least-squares problem with orthonormality constraints

Given $y_1,\dots, y_n \in {\Bbb R}$ , $w \in {\Bbb R}^d$ , and $x_1,\dots, x_n \in {\Bbb R}^D$ , how do we solve the following optimization problem \begin{align} \min_A \,\, \sum_{i=1}^n \left( y_i - w^T A^T x_i \right)^2\\ \text{subject to} \qquad…
gmravi
  • 386
5
votes
1 answer

Cell structure on Stiefel manifolds

Denote by $V_{n,k}$ the set of $k$-frames in $\mathbb{R}^n$. This is a Stiefel manifold. I am trying to understand the following statement of the cellular structure (from Mosher & Tangora's book page 42) By a normal cell of $V_{n,k}$ we mean a cell…
Juan S
  • 10,596
5
votes
1 answer

Equation for geodesic in manifold of orthogonal matrices

From Geodesic on Stiefel manifold , a geodesic (under the canonical metric) on the manifold of orthogonal matrices can be expressed as $$Y(t) = Q e^{Xt} I$$ for some matrices $Q$ and $X$ . Does it follow that the geodesic between two orthogonal…
opt_learn
  • 53
5
votes
0 answers

Stiefel manifold is a manifold

We consider the real Stiefel manifold $$V_k (\mathbb{R}^n) := \left\{ (v_1, v_2, \dots, v_k) \in S^{n-1} \times \cdots \times S^{n-1} \mid \langle v_i, v_j \rangle = \delta_{ij} \right\}$$ I want to know how to show that $V_k(\mathbb{R}^n)$ is a…
user267839
  • 10,221
5
votes
1 answer

Fiber bundle on Stiefel manifold

Let $V_{n}(\mathbb{C}^{k})$ the Stiefel manifold of $n$-frame in $\mathbb{C}^{k}$. We can see $V_{n}(\mathbb{C}^{k})$ as a subset of $n$ copies of the cartesian product $S^{2k-1} \times \cdots \times S^{2K-1}$. So we have a bundle $…
ArthurStuart
  • 51
4
votes
0 answers

Riemannian submersion from Stiefel to Grassmann manifold having totally geodesic fibres

Consider the Lie group of unitary matrices endowed with its bi-invariant metric $(U(n), g)$ , where $g(X, Y) := \text{Tr}(Y^\dagger X)$ for any two tangent vectors $X, Y$ . Let us also consider the complex Stiefel manifold $V_k(\mathbb{C}^n) \simeq…
mathusername
  • 189
4
votes
1 answer

Map between Stiefel manifold and the Grassmannian

I'm working on problem 2-7 in Lee's introduction to Riemannian Manifolds and am having trouble on part (b): Let $V_k(\mathbb{R}^n)$ be the Stiefel manifold and $G_k(\mathbb{R}^n)$ denote the Grassmannian. I want to show that the map $$ \pi :…
ABC
  • 379
4
votes
5 answers

Maximize $\mathrm{tr}(Q^TCQ)$ subject to $Q^TQ=I$

Let $C \in \mathbb{R}^{d \times d}$ be symmetric, and $$Q = \begin{bmatrix} \vert & \vert & & \vert \\ q_1 & q_2 & \dots & q_K \\ \vert & \vert & & \vert \end{bmatrix} \in \mathbb{R}^{d\times K}$$ where $d \geq K$ . Using Lagrange multipliers,…
abcd
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