# Questions tagged [stiefel-manifolds]

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

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### What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?

Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
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### differentiate a mtrix function with respect to a matrix

I want to differentiate $f_{\theta}(\nabla_{\theta}(X))$ with respect to $\theta$, where $\theta$ is orthogonal $n \times p$ matrix (lies on Stiefel manifold) and $\nabla_{\theta}(X)$ is the ...
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Let $C \in \mathbb{R}^{d \times d}$ be symmetric, and $$Q = \begin{bmatrix} \vert & \vert & & \vert \\ q_1 & q_2 & \dots & q_K \\ \vert & \vert &... • 279 0 votes 1 answer 374 views ### Stiefel manifold is homeomorphic to O(n)/O(n - k) I have a question on Is the Stiefel manifold V_k(\mathbb{R}^n) homeomorphic to O(n)/O(n-k)? Why is the described map bijective? The statement 'A \sim B \iff AB^T \in O(n - k) so V_k(\mathbb{R}^... • 2,554 2 votes 1 answer 179 views ### Optimizing Trace(Q^TZ) subject to Q^TQ=I Let Z \in \mathbb{R}^{m \times n} be a tall matrix (m > n). Solve the following optimization problem in Q \in \mathbb{R}^{m \times n}$$\begin{array}{ll} \text{maximize} & \mbox{Tr} \left(...
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Grassmann manifold $G_{k,n}$ is the set of k-dimensional subspaces of $\mathbb{R^n}$. Let’s consider the set of k-frames $V_{k,n}$. I want to show that $$G \to V_{k,n} \to^{\pi} G_{k,n}$$ can be ...