# Geometry of the Symplectic Stiefel Manifold Endowed with the Euclidean Metric

@inproceedings{Gao2021GeometryOT, title={Geometry of the Symplectic Stiefel Manifold Endowed with the Euclidean Metric}, author={Bin Gao and Nguyen Thanh Son and Pierre-Antoine Absil and Tatjana Stykel}, booktitle={GSI}, year={2021} }

The symplectic Stiefel manifold, denoted by Sp(2p, 2n), is the set of linear symplectic maps between the standard symplectic spaces R and R. When p = n, it reduces to the well-known set of 2n × 2n symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space R. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel…

## 2 Citations

The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications

- Mathematics, Computer ScienceArXiv
- 2021

This work studies the real symplectic Stiefel manifold with the goal of providing theory and matrixbased numerical tools fit for basic data processing and treats the ‘nearest symplectic matrix’ problem and the problem of optimal data representation via a low-rank symplectic subspace.

Constraint optimization and SU(N) quantum control landscapes

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

We develop the embedded gradient vector field method, introduced in [8, 9], for the case of the special unitary group SU(N) regarded as a constraint submanifold of the unitary group U(N) . The…

## References

SHOWING 1-10 OF 22 REFERENCES

Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices

- MathematicsSIAM J. Matrix Anal. Appl.
- 2011

The minimal-distance problem investigated in this paper relies on a suitable notion of distance—induced by the Frobenius norm—as opposed to the natural pseudodistance that corresponds to the pseudo-Riemannian metric that the real-symplectic group is endowed with.

Symplectic eigenvalue problem via trace minimization and Riemannian optimization

- Mathematics
- 2021

We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson’s theorem. It is…

Riemannian Optimization on the Symplectic Stiefel Manifold

- MathematicsSIAM J. Optim.
- 2021

This paper considers a novel Riemannian metric akin to the canonical metric of the (standard) Stiefel manifold, and develops two types of search strategies: one is based on quasi-geodesic curves, and the other one on the symplectic Cayley transform.

A Riemannian‐steepest‐descent approach for optimization on the real symplectic group

- Mathematics
- 2018

In this paper, we first give the geodesic in closed form on the real symplectic group endowed with a Riemannian metric and then study a geodesic‐based Riemannian‐steepest‐descent approach to compute…

Critical Landscape Topology for Optimization on the Symplectic Group

- Mathematics
- 2010

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over…

Symplectic Model Reduction of Hamiltonian Systems

- MathematicsSIAM J. Sci. Comput.
- 2016

This paper introduces three algorithms for PSD, which are based upon the cotangent lift, complex singular value decomposition, and nonlinear programming, and shows how PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems.

Derivatives of symplectic eigenvalues and a Lidskii type theorem

- MathematicsCanadian Journal of Mathematics
- 2020

Abstract Associated with every
$2n\times 2n$
real positive definite matrix
$A,$
there exist n positive numbers called the symplectic eigenvalues of
$A,$
and a basis of
$\mathbb {R}^{2n}$
…

On symplectic eigenvalues of positive definite matrices

- Mathematics
- 2015

If A is a 2n × 2n real positive definite matrix, then there exists a symplectic matrix M such that MTAM=DOOD where D = diag(d1(A), …, dn(A)) is a diagonal matrix with positive diagonal entries, which…